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DairyMod and the SGS Pasture Model
A mathematical description of the biophysical model structure
Ian R Johnson
Published by IMJ Consultants
PO Box 182
Dorrigo
NSW 2453
Australia
www.imj.com.au
Originally published November 2005
This version published 26 April 2016
ยฉCopyright, IMJ Consultants Pty Ltd, 2005-2016. All rights reserved.
Citation:
Johnson IR (2016). DairyMod and the SGS Pasture Model: a mathematical description of the biophysical
model structure. IMJ Consultants, Dorrigo, NSW, Australia.
DairyMod and the SGS Pasture Model
1 Background to biophysical modelling ................................................................................... 1
1.1 Introduction ........................................................................................................................................ 1
1.2 Plant, crop and pasture modelling ..................................................................................................... 2
1.2.1 Hierarchical systems .................................................................................................................. 2
1.2.2 Types of models ......................................................................................................................... 2
1.3 Background mathematical functions .................................................................................................. 3
1.3.1 Rectangular hyperbola .............................................................................................................. 3
1.3.2 Non-rectangular hyperbola ....................................................................................................... 5
1.3.3 Switch functions ........................................................................................................................ 7
1.3.4 CO2 response function ............................................................................................................... 7
1.3.5 Temperature response functions .............................................................................................. 9
1.3.6 Gompertz growth equation ..................................................................................................... 13
1.4 Eulerโs method for solving differential equations ............................................................................ 16
1.5 Plant composition components ........................................................................................................ 17
1.6 Atmospheric composition ................................................................................................................ 18
1.6.1 CO2 concentration ................................................................................................................... 20
1.6.2 Water vapour ........................................................................................................................... 21
1.7 Final comments ................................................................................................................................ 22
1.8 References ........................................................................................................................................ 23
2 Climate ............................................................................................................................. 24
2.1 Introduction ...................................................................................................................................... 24
2.2 Rainfall .............................................................................................................................................. 24
2.3 Temperature ..................................................................................................................................... 25
2.4 Radiation ........................................................................................................................................... 25
2.4.1 Black body radiation ................................................................................................................ 26
2.4.2 Non-black body and grey body radiation ................................................................................ 27
2.4.3 Radiation energy for photosynthesis: PAR and PPF ............................................................... 27
2.4.4 Radiation units and terminology ............................................................................................. 28
2.4.5 Canopy light interception and attenuation ............................................................................. 28
2.4.6 Clear-sky solar radiation and daylength .................................................................................. 31
2.4.7 Net radiation ............................................................................................................................ 36
2.5 Final comments ................................................................................................................................ 37
2.6 References ........................................................................................................................................ 37
3 Pasture and Crop Growth .................................................................................................. 39
3.1 Introduction ...................................................................................................................................... 39
3.2 Transpiration and the influence of water stress .............................................................................. 40
3.3 Canopy photosynthesis .................................................................................................................... 41
3.3.1 Units ......................................................................................................................................... 42
3.3.2 Leaf gross photosynthesis ....................................................................................................... 42
3.3.3 Instantaneous canopy gross photosynthesis........................................................................... 47
3.3.4 Daily canopy gross photosynthesis .......................................................................................... 48
3.3.5 Daily canopy respiration rate .................................................................................................. 49
3.3.6 Daily carbon fixation ................................................................................................................ 51
3.3.7 Influence of temperature extremes on photosynthesis .......................................................... 51
3.4 Root distribution ............................................................................................................................... 53
3.5 Nitrogen remobilisation, uptake, and fixation ................................................................................. 53
3.6 Pasture growth, senescence and development ............................................................................... 54
3.6.1 Shoot:root partitioning ............................................................................................................ 55
3.6.2 Leaf:sheath partitioning and leaf growth ................................................................................ 56
3.6.3 Growth dynamics ..................................................................................................................... 56
3.6.4 Influence of temperature on growth dynamics ...................................................................... 58
3.7 Mixed swards .................................................................................................................................... 58
3.7.1 Light interception and attenuation ......................................................................................... 58
3.7.2 Root distribution ...................................................................................................................... 60
3.8 Crop growth ...................................................................................................................................... 61
3.1.1 Cereals ..................................................................................................................................... 61
3.1.2 Brassicas .................................................................................................................................. 64
3.9 Concluding remarks .......................................................................................................................... 64
3.10 References ........................................................................................................................................ 65
4 Water dynamics ................................................................................................................ 67
4.1 Introduction ...................................................................................................................................... 67
4.2 Potential transpiration ..................................................................................................................... 67
4.2.1 Penman-Monteith equation .................................................................................................... 68
4.3 Potential daily evaporation .............................................................................................................. 70
4.4 Soil water infiltration and redistribution .......................................................................................... 70
4.5 Runoff ............................................................................................................................................... 72
4.6 Evaporation ...................................................................................................................................... 73
4.6.1 Canopy ..................................................................................................................................... 73
4.6.2 Litter ........................................................................................................................................ 74
4.6.3 Soil ........................................................................................................................................... 74
4.7 Concluding remarks .......................................................................................................................... 75
4.8 References ........................................................................................................................................ 75
5 Soil organic matter and nitrogen dynamics ........................................................................ 76
5.1 Introduction ...................................................................................................................................... 76
5.2 Organic matter dynamics ................................................................................................................. 77
5.2.1 Overview .................................................................................................................................. 77
5.2.2 Organic matter turnover ......................................................................................................... 78
5.2.3 Effects of water and temperature ........................................................................................... 80
5.2.4 Influence of inputs on organic matter dynamics ..................................................................... 81
5.2.5 Half-life and mean-residence time .......................................................................................... 81
5.2.6 Initialization ............................................................................................................................. 82
5.2.7 Illustration................................................................................................................................ 83
5.2.8 Surface litter and dung ............................................................................................................ 84
5.3 Inorganic nutrient dynamics ............................................................................................................. 84
5.3.1 Overview .................................................................................................................................. 84
5.3.2 Nitrogen inputs ........................................................................................................................ 84
5.3.3 Nitrification of ammonium ...................................................................................................... 84
5.3.4 Denitrification of nitrate ......................................................................................................... 86
5.3.5 Volatilization of ammonium .................................................................................................... 89
5.3.6 Nutrient adsorption ................................................................................................................. 89
5.3.7 Nutrient leaching ..................................................................................................................... 92
5.4 Concluding remarks .......................................................................................................................... 92
5.5 References ........................................................................................................................................ 92
6 Animal growth and metabolism ......................................................................................... 94
6.1 Introduction ...................................................................................................................................... 94
6.2 Body composition during growth ..................................................................................................... 95
6.3 Growth and energy dynamics ........................................................................................................... 97
6.4 Model solution in relation to available energy .............................................................................. 100
6.4.1 ๐ฌ๐๐ exceeds requirements for normal growth ..................................................................... 101
6.4.2 ๐ฌ๐๐ is between maintenance requirement and normal growth requirement ..................... 101
6.4.3 ๐ฌ๐๐ is less than maintenance requirement .......................................................................... 101
6.5 Illustrations of animal growth dynamics ........................................................................................ 102
6.6 Pregnancy and lactation ................................................................................................................. 108
6.6.1 Pregnancy .............................................................................................................................. 108
6.6.2 Lactation ................................................................................................................................ 110
6.7 Animal intake .................................................................................................................................. 111
6.7.1 Potential intake ..................................................................................................................... 111
6.7.2 Intake in relation to feed composition .................................................................................. 113
6.7.3 Pasture intake ........................................................................................................................ 114
6.7.4 Supplement intake................................................................................................................. 115
6.7.5 Substitution ........................................................................................................................... 115
6.8 Metabolisable energy and nitrogen dynamics ............................................................................... 116
6.9 Growth dynamics in response to metabolisable energy ................................................................ 118
6.10 Final comments .............................................................................................................................. 120
6.11 References ...................................................................................................................................... 121
7 DairyMod management ................................................................................................... 123
7.1 Introduction .................................................................................................................................... 123
7.2 Stock ............................................................................................................................................... 123
7.3 Supplement .................................................................................................................................... 123
7.4 Feed management .......................................................................................................................... 123
7.5 Single paddock management ......................................................................................................... 124
7.5.1 Cutting simulation ................................................................................................................. 125
7.5.2 Grazing simulation ................................................................................................................. 125
7.5.3 Variable stock density ............................................................................................................ 125
7.5.4 Rotational grazing by pasture dry weight .............................................................................. 126
7.5.5 Rotational grazing at fixed time interval ............................................................................... 126
7.5.6 Rotational grazing by feed on offer and days on paddock .................................................... 126
7.5.7 Rotational grazing by date ..................................................................................................... 126
7.6 Multiple paddock management ..................................................................................................... 126
7.6.1 Fixed time rotation ................................................................................................................ 126
7.6.2 Rotate in relation to pasture dry weight ............................................................................... 126
7.6.3 Rotate in relation to leaf stage and pasture dry weight ........................................................ 127
7.6.4 Cutting rules .......................................................................................................................... 127
7.6.5 Rotational grazing phases ...................................................................................................... 127
7.6.6 Crops ...................................................................................................................................... 127
7.7 Nutrient removal ............................................................................................................................ 128
7.8 Nitrogen fertilizer ........................................................................................................................... 129
7.9 Irrigation ......................................................................................................................................... 129
7.10 Final comments .............................................................................................................................. 130
Chapter 1: Background to biophysical modelling 1
1 Background to biophysical modelling
1.1 Introduction
The SGS Pasture Model and DairyMod are biophysical pasture simulation models, with a common
underlying structure. There are modules for pasture growth and utilization by grazing animals, animal
physiology including animal production, water and nutrient dynamics, as well as a range of options for
pasture management, irrigation and fertilizer application. The principal features of the individual modules
are:
The pasture growth module includes calculations of light interception and photosynthesis; growth
and maintenance respiration, nutrient uptake and nitrogen fixation, partitioning of new growth
into the various plant parts, development, tissue turnover and senescence, and the influence of
atmospheric CO2 on growth. The model allows up to five pasture species in any simulation, which
can be annual or perennial, C3 or C4, as well as legumes.
The water module accounts for rainfall and irrigation inputs that can be intercepted by the canopy,
surface litter or soil. The required hydraulic soil parameters are saturated hydraulic conductivity,
bulk density, field capacity or drained upper limit, wilting point and air-dry water content.
(Saturated water content is calculated from bulk density.)
Different soil physical properties can be defined through the soil profile. The nutrient module
incorporates the dynamics of inorganic nitrogen (including leaching) and soil organic matter. The
soil organic matter module gives a complete description of soil carbon in the system and its
response to factors such as climate variation and management. In addition, gaseous losses of
nitrogen through volatilization and denitrification are included.
The animal module has a sound treatment of animal intake and metabolism including growth,
maintenance, pregnancy and lactation. There are options to select sheep (wethers or ewes with
lambs), cattle (steers or beef cows with calves), and dairy cows. Wool growth is included for sheep.
The farm management module describes the movement of stock around paddocks as well as
strategies for conserving forage, and incorporates a wide range of rotational grazing management
strategies that are used in practice. There are options for single- and multi-paddock simulations
that can each be defined independently to represent spatial variation in soil types, nutrient status,
pasture species, fertilizer and irrigation management.
The model has been applied to a range of research questions, which include comparisons with
experimental data from several disparate geographical locations for a range of pasture species, as well as
addressing important questions such as climate variability, drought, business risk, and the impacts of
climate change and carbon mitigation strategies. The biophysical structure of the model means it is well
suited to be developed to explore issues such as new management strategies or plant characteristics, as
well as the environmental impacts of likely future climate change scenarios.
This book provides a complete mathematical description of the model. Before embarking on the model
description, this first chapter provides a detailed background to the mathematical methods that are used in
the model, as well as some general topics of importance. For a further discussion of modelling techniques
see Thornley and Johnson (2000) and Thornley and France (2007).
DairyMod and the SGS Pasture Model documentation 2
1.2 Plant, crop and pasture modelling
In recent years, models have become an integral component of the plant, crop and pasture sciences. They
have a wide variety of uses in many aspects of agricultural management, such as irrigation scheduling and
pasture management. Equally so, they play an important role in plant and crop research: for example,
models have helped identify the growth and maintenance components of plant respiration. This interest in
modelling has had many benefits and has provided a means of integrating concepts from many different
branches of science.
When constructing models it is important that model design meets the objectives of the project. It is
therefore important to consider the different types of model and how these relate to each other. The main
types used in this model are mechanistic and empirical, as discussed in Section 1.2.2 below. Before
proceeding, it is worth noting that it is unlikely that one model of a particular process will suit all likely
modelling objectives.
1.2.1 Hierarchical systems
Plant biology, and biology in general, has many organizational levels. In physics and chemistry there is a
clear distinction between moving from atomic and molecular behaviour to that of liquids and solids, in
biology there are several levels that can be considered. This range of different levels gives rise to the great
diversity of the biological world. A typical hierarchical scheme for the plant sciences is:
โฆ โฆ
โฆ landscape
๐ + 1 crop or pasture
๐ plant
๐ โ 1 organs
โฆ tissues
โฆ cells
โฆ organelles
โฆ macromolecules
โฆ molecules and atoms
The model described here focuses on processes primarily at levels ๐ + 1, ๐, ๐ โ 1.
The principal features of this hierarchical system are:
1. Each level has its own language. For example, crop yield has little meaning at the cell level.
2. Each level is an integration of items at lower levels, so that the response of the system at one level
can be related to responses at lower levels. For example, canopy photosynthesis is calculated in
terms of the sum of the photosynthesis of the leaves in the plants that make up the canopy.
Other features of this system are discussed in Thornley and Johnson (2000) and Thornley and France
(2007).
1.2.2 Types of models
Models can be divided into several categories, with perhaps the most widely used being mechanistic and
empirical, deterministic and stochastic. The difference between deterministic and stochastic models is that
deterministic models predict a precise value for a variable of the system, whereas stochastic models involve
statistical variation. Both have their value, but in the present model the statistical features of the
behaviour of the system are not considered.
Chapter 1: Background to biophysical modelling 3
Empirical models
Empirical models are curves that are used to describe patterns of behaviour or to summarize sets of data
but do not involve details of the underlying scientific basis of the system. An example of empirical
modelling is the use of growth functions to describe animal weight during growth. Generally, an empirical
model describes the response of the system at a single level in the hierarchical structure mentioned in the
previous section.
Although empirical models are usually curves that can be fitted to experimental data, and that display the
general expected characteristics of the response, they are much more useful if the curves have readily
interpreted parameters. For example, the temperature response functions discussed in Section 1.3.5
below are used to describe the influence of temperature on various processes in the model. These
response functions are empirical but are formulated with minimum, optimum and maximum temperature
parameters for the processes, which makes them simple to apply.
Mechanistic models
Mechanistic models are constructed from descriptions of the underlying processes involved in the system
being studied, and these descriptions are quite often empirical (or semi-empirical). They generally operate
between two or three levels in the hierarchical structure discussed in the previous section. For example,
canopy gross photosynthesis can be defined in terms of an equation describing single leaf photosynthesis in
response to light and another equation describing light attenuation through the canopy. Ideally, each of
these sub-models will have parameters that have some biophysical interpretation, but they may not be
founded on detailed mechanisms.
The complexity of mechanistic models will increase as the range of processes used to build that model
increases, or if greater detail is used to describe these processes. The complexity at which a model is
developed is therefore subject to some degree of choice. The greater the detail, the more complex the
model. It is important that the complexity of the model suits the objectives of the system being
investigated and this means that there is generally no single model of a biological process that suits all
purposes.
The distinction between mechanistic and empirical models is not always clear. For example, the equation
used to describe single leaf gross photosynthesis in the model, the non-rectangular hyperbola which is
discussed in detail later, can be derived from a very simple model of leaf photosynthesis. However, the
underlying model contains such broad assumptions that it does not really encapsulate the biochemical
details of leaf photosynthesis. In this case, we can regard the equation as semi-empirical, recognising that
it has the desired behaviour but with a limited biophysical basis.
1.3 Background mathematical functions
Some background calculations are now presented. These are used in the later model descriptions.
1.3.1 Rectangular hyperbola
The simple rectangular hyperbola (RH) can be derived from basic concepts of enzyme kinetics. It is
generally presented in one of two forms:
m
m
xyy
x y (1.1)
or
DairyMod and the SGS Pasture Model documentation 4
mx
y yx K
(1.2)
although the symbol ๐ฃ for the speed of the reaction, and ๐ for substrate concentration are often used
instead of ๐ฅ and ๐ฆ. In both equations, ๐ฆ๐ is the asymptotic value of ๐ฆ as ๐ฅ โ โ; with eqn (1.1), ๐ผ is the
initial slope of the curve, while for eqn (1.2) ๐ฆ takes half its maximum value when ๐ฅ = ๐พ, that is ๐ฆ(๐ฅ =
๐พ) = ๐ฆ๐ 2โ . These two forms of the RH are mathematically equivalent and it is readily shown that
myK (1.3)
The rectangular hyperbola of the form (1.2) is often referred to as the Michaelis-Menten equation due to
their early application of this equation to enzyme kinetics in 1913.
The form of the equation that is used depends on the particular application โ sometimes it is convenient to
prescribe the initial slope of the response, ๐ผ, while in other cases the value of ๐ฅ for half-maximal response,
๐พ, is more convenient.
The equation is referred to as a rectangular hyperbola since it has two asymptotes that are at right angles
to each other. These are:
my y (1.4)
and
myx
or x K (1.5)
In practice, only positive values of ๐ฅ are used. The RH is illustrated in Fig. 1.1.
Figure 1.1: Rectangular hyperbola (blue lines), eqn (1.1) or (1.2), and the asymptotes given by
eqns (1.4) and (1.5) (black dashed lines). The solid blue line is the part of the equation that is
generally used in biological models.
The RH is also shown in Fig. 1.2 where now only the part of the curve that is biologically meaningful is
shown, along with the key equation parameters. While the RH is a simple curve to work with, and the
parameters have biological meaning, it is limited in that it generally approaches the asymptote quite slowly.
The more general non-rectangular hyperbola that is discussed in the next section overcomes this limitation.
y
x
Chapter 1: Background to biophysical modelling 5
Figure 1.2: Rectangular hyperbola (blue line) with the key parameters as indicated.
See text for details.
1.3.2 Non-rectangular hyperbola
The non-rectangular hyperbola (NRH) is a useful generic equation that is widely used to describe the leaf
photosynthetic response to irradiance (see Chapter 3). It will also be used here for the CO2 response.
Mathematically, it is a modification of the rectangular hyperbola (RH) discussed in the previous section that
has an extra parameter and where the asymptotes are now not perpendicular to each other. An overview
of the equation is given here and for more detail, see Thornley and Johnson (2000) or Thornley and France
(2007). Adding a quadratic equation to the RH, the NRH equation can be written
2 0 m my x y y xy (1.6)
For ๐ = 0 it reduces to the RH, eqn (1.1).
The solutions to eqn (1.6) are
1 2
214
2
m m my x y x y y x (1.7)
Note that for this equation to have two real solutions, it is necessary that
2
4 m my x x y (1.8)
for all values of ๐ฅ. For ๐ฅ โฅ 0 this requires
2
4
m
m
x y
y x (1.9)
It is easy to show that the right-hand side of this equation takes its minimum value of 1 when ๐ผ๐ฅ = ๐ฆ๐, so
that the required constraint is
1 (1.10)
Before looking at these solutions given by (1.7), note that eqn (1.6) can be factorized to give
21 1 m m my y y x y y (1.11)
from which it can be seen that there are two asymptotes at
y
x
ym
K
ym /2
0
DairyMod and the SGS Pasture Model documentation 6
my y and 1
mx yy (1.12)
The solutions to eqn (1.6) as given by (1.7), along with the asymptotes (1.11) are illustrated in Fig. 1.3.
Figure 1.3: Non-rectangular hyperbola (blue lines), eqns (1.6) and (1.7), and the asymptotes
given by eqns (1.12) (black dashed lines). The solid blue line is the part of the equation that is
generally used in biological models.
In biological models, and all the present applications, the lower solution in (1.7) is used, with ๐ฅ โฅ 0, that is
1 2
214 0
2
,m m my x y x y y x x (1.13)
This is illustrated with the solid line in Fig. 1.3 and for a range of ๐ values in Fig. 1.4. This is a powerful,
versatile equation that is easy to work with. The three parameters each control the key aspects of the
response: the initial slope, curvature and asymptote. This is the form of the equation that is used to
describe the light response for leaf gross photosynthesis. It is also used in the CO2 response function.
Figure 1.4: Non-rectangular hyperbola, eqn (1.13) for ๐ increasing from 0 (lower line,) to 1
(upper line). The initial slope is ๐ผ and the asymptote is ๐ฆ๐.
y
x
y
x 0
ym
๐ = 0
๐ = 1
Chapter 1: Background to biophysical modelling 7
1.3.3 Switch functions
It is sometimes useful to have expressions to define โswitch-onโ or โswitch-offโ behaviour. Simple equations
for this are
n
on m n n
xy y
x K (1.14)
which is quite similar to the rectangular hyperbola, and
n
off m n n
Ky y
x K (1.15)
Both of these equations, which are illustrated in Fig. 1.5, take the value ๐ฆ๐ 2โ when ๐ฅ = ๐พ.
Figure 1.5: โSwitch-onโ (left) and โswitch-offโ (right) functions given by eqns (1.14) and (1.15),
with ๐ฆ๐ = 1, ๐พ = 1 and ๐ as indicated
These functions are not actually used in the present model, but are presented here as simple extensions of
the rectangular hyperbola that may be of use in other modelling exercises.
1.3.4 CO2 response function
The NRH will be used in the treatment of the photosynthetic response to CO2. However, it is convenient to
re-cast the equation. First consider the general equation given by
1 2
214
2
, , ,C C m C m C mf C C f C f f C (1.16)
which is of the form eqn (1.13), where ๐ถ is atmospheric CO2 concentration, ๐ฝ is the initial slope, ๐
(0 โค ๐ โค 1) the curvature and ๐๐ถ,๐ the asymptote.
In order to assist with parameterization, the function is constrained to take the value unity at ambient CO2
and ๐ at double ambient, so that
1
2
C amb
C amb
f C C
f C C (1.17)
which are the values at ambient and double ambient CO2 concentration, where ๐ถ๐๐๐ is the ambient
atmospheric CO2 concentration, taken to be ๐ถ๐๐๐ = 380 ๐mol molโ1, eqn (1.91) below.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
y on
x
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
y off
x
n=1
n=2
n=3
n=4
DairyMod and the SGS Pasture Model documentation 8
For example, consider this equation as used for leaf gross photosynthesis at saturating irradiance. If
๐ = 1.5 and ๐๐ถ,๐ = 2 then the photosynthetic rate increases by 50% when CO2 is double ambient, and is
increased by 100% at saturating CO2. It is now necessary to calculate the appropriate values of ๐ฝ and ๐ in
eqn (1.16) in order to satisfy (1.17). After some algebra, it can be shown that
2
1 2
1 2
, , ,
, ,
C m C m C m
C m C m
f f f
f f (1.18)
and
2
,
,
C m
amb C m
f
C f (1.19)
so that ๐ฝ and ๐ are evaluated in terms of ๐ and ๐๐ถ,๐. Care must be taken to ensure that values for ๐ and
๐๐ถ,๐ are selected such that 0 โค ๐ โค 1 and ๐ฝ > 0. To do so, note that
1 when ,C mf ; and 0 when 2 ,C mf (1.20)
Since ๐ > 1, it then follows that the required constraint is
2
,C mf (1.21)
This is checked in the program. Note that with the default values
1 5 2 ,. , C mf (1.22)
it follows that
0 0032 0 8 . , . (1.23)
The function is illustrated in Fig. 1.6 where ambient, double ambient, and the asymptote are also shown.
Figure 1.6: Generic CO2 response function ๐๐ถ, eqn (1.16), subject to (1.17), (1.18), (1.19)with
๐ถ๐๐๐ = 380ppm, ๐๐ถ(๐ถ = ๐ถ๐๐๐) = 1, ๐๐ถ(๐ถ = 2๐ถ๐๐๐) = 1.5, ๐๐ถ(๐ถ โ โ) = 2.
0.0
0.5
1.0
1.5
2.0
0 500 1000 1500 2000
CO
2 f
un
ctio
n, f
c
CO2 concentration, ppm
Chapter 1: Background to biophysical modelling 9
1.3.5 Temperature response functions
Two forms of temperature response are used in the model โ either with or without a temperature
optimum. For more details see Johnson and Thornley (1985), Thornley and Johnson (2000), Thornley and
France (2007).
Temperature response without an optimum
The simplest equation to use is the so-called ๐10, which is given by
10
10
rT T
rk k Q (1.24)
where ๐ is the reaction rate, ๐ is temperature, ๐๐ is a reference temperature, taken to be
rT 20โ (1.25)
๐๐ is the value of ๐ at the reference temperature ๐๐, and ๐10 is the temperature coefficient. According to
this equation,
10
10
k TQ
k T (1.26)
for all values of ๐, so that the reaction rate increases by a factor ๐10 for every 10โ increase in
temperature. ๐10 is typically of order 1.5 to 2 for most practical applications.
An alternative equation that is sometimes used is the Arrhenius equation, which is defined by
aE RTk Ae (1.27)
where ๐ด is a rate parameter with the same dimensions as ๐, ๐ธ๐ (J mol-1) is the activation energy, ๐ (K) is the
absolute temperature, and ๐ = 8.3145 J K-1 mol-1 is the gas constant. A derivation of eqn (1.27) can be
found in Johnson and Thornley (1985) or Thornley and Johnson (2000).
It is convenient to normalize (1.27) so that it takes a reference value at the reference temperature ๐๐,
which requires
293 15
.aE RrA k e (1.28)
where the factor 293.15 is 20โ converted to absolute degrees (K). Equation (1.27) is now written
1 1
exp ar
r
Ek k
R T T (1.29)
so that
r rk k T T (1.30)
In practice, the activation energy, ๐ธ๐ is treated as an empirical parameter to fit to data. This can be
compared to the ๐10 equation, eqn (1.24), by using the fact that they both take the value ๐๐ at the
reference temperature, ๐๐, and then equating them at 30โ to give
10 10293 15 303 15
8 3145 74 13010
. .. ln , lnaE Q Q (1.31)
With typical values of 1.5 and 2 for ๐10, the corresponding ๐ธ๐ values are
DairyMod and the SGS Pasture Model documentation 10
1
10
110
1 5 29 960
2 51 216
. ; , J mol
; , J mol
a
a
Q E
Q E (1.32)
The ๐10 and Arrhenius equations are illustrated in Fig. 1.7 for these parameter values. It can be seen that
the responses are virtually identical over a practical temperature range.
Figure 1.7. ๐10 equation, blue lines, and Arrhenius equation, dashed red lines, for the ๐10
values as indicated and activation energies given by eqn (1.32). The two equations give
virtually identical responses.
Given the close similarity between the two equations, the choice here is to use the ๐10 approach since the
๐10 parameter is intuitive to work with and is simple to relate to data. Furthermore, the Arrhenius
equation is based on a single chemical reaction, whereas processes such as plant respiration involve
sequences of many reactions which may have different individual energy characteristics.
Temperature response with an optimum
Temperature responses with a temperature optimum are more complex to deal with than those without
that were considered above. The scheme for the Arrhenius equation can be generalized to generate a
response function that has a maximum, and the resulting equation is
1
exp
exp
aA E RTk
S R H RT (1.33)
where, again, ๐ด is a rate constant, ๐ธ๐ (J mol-1) is the activation energy, ๐ (K) is the absolute temperature,
and ๐ is the gas constant. The additional parameters are โ๐ (J K-1 mol-1) which is an entropy term, and โ๐ป
(J mol-1) which is an enthalpy term. A derivation of eqn (1.33) is given in Johnson and Thornley (1985) or
Thornley and Johnson (2000). Equation (1.33) is illustrated in Fig. 1.8.
0
1
2
3
4
5
0 10 20 30 40 50
Re
acti
on
rat
e, k
Temperature, T (ยฐC)
Q10=1.5
Q10=2
Chapter 1: Background to biophysical modelling 11
Figure 1.8: Temperature response function, eqn (1.33).
Blue line: ๐ด = 5.33 ร 1010, ๐ธ๐ ๐ โ = 7.5 ร 103 K, โ๐ ๐ โ = 48, โ๐ป ๐ โ = 1.5 ร 104 K.
Red line: ๐ด = 2.5 ร 1010, ๐ธ๐ ๐ โ = 1.5 ร 103 K, โ๐ ๐ โ = 81, โ๐ป ๐ โ = 2.5 ร 104 K
While parameter values can be selected in eqn (1.33) to describe temperature responses, it is quite
complex to vary the parameters routinely to adjust the details of the curve. As for the Arrhenius equation
discussed above, eqn (1.27), the underlying scheme that leads to eqn (1.33) involves an idealized single
enzyme-substrate reaction where the enzyme can exist in either an active or inactive state. There is little
theoretical justification in using this scheme for the sequence of reactions that occur in photosynthesis. It
should be noted that variations to eqn (1.33) can be derived.
For the present purposes, a simpler empirical temperature response function is used. Following Thornley
(1998), Thornley and France (2007), consider the temperature response function given by
qmn mx
Tr mn mx r
T T T Tf T
T T T T (1.34)
where ๐๐๐ and ๐๐๐ฅ are the minimum and maximum temperatures such that
0 T mn T mxf T f T (1.35)
๐ โฅ 1 is a curvature parameter, and ๐๐ is a reference temperature with
1T rf T (1.36)
This equation has a maximum value at
1
mn mxopt
T qTT
q (1.37)
from which
1
opt mn
mx
q T TT
q (1.38)
Equation (1.38) can be used in (1.34) to eliminate ๐๐๐ฅ, giving
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 10 20 30 40 50
Re
acti
on
rat
e, k
Temperature, T (ยฐC)
DairyMod and the SGS Pasture Model documentation 12
0
1
1
0
,
,
mn
qopt mnmn
T mn mxr mn opt mn r
mx
T T
q T T qTT Tf T T T T
T T q T T qT
T T
(1.39)
where ๐๐๐ฅ is given by eqn (1.38). This ๐๐๐, ๐๐๐๐ก equation describes the temperature response in terms of
the minimum and optimum temperatures, as well as the curvature coefficient ๐. Alternatively, if it was
more convenient, ๐ could be derived from eqn (1.37) to give an equation in terms of the minimum,
optimum and maximum temperatures. This is not presented here, and eqn (1.39) will be used.
In applying this function, the constraint
r optT T (1.40)
should be applied. While this is not absolutely necessary, it does ensure sensible behaviour of eqn (1.39).
The function is illustrated in Fig. 1.9.
Figure 1.9. Generic temperature function, eqn (1.39). Parameters are: ๐๐ = 20โ, ๐๐๐ = 5โ,
๐๐๐๐ก = 25โ, ๐ as indicated. Note that ๐(๐ = ๐๐) = 1.
This equation is versatile and simple to use, having easily interpreted parameter values, and will be used for
temperature responses that have an optimum. It should be noted that when ๐ = 1 the temperature
response is symmetric around the optimum temperature. This rarely happens in biological processes and
values of ๐ in the range 2 to 3 are generally more appropriate.
For situations where a temperature function without an optimum is required, eqn (1.39) is modified to be
0
1
1
1
,
,
mn
qopt mnmn
T mn optr mn opt mn r
qopt mn opt mn
optr mn opt mn r
T T
q T T qTT Tf T T T T
T T q T T qT
T T T TT T
T T q T T qT
(1.41)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0 10 20 30 40
Tem
per
atu
re fu
nct
ion
, f(T
)
Temperature, T (ยฐC)
q=1
q=2
q=3
Chapter 1: Background to biophysical modelling 13
This equation is illustrated in Fig. 1.10 for the same parameter values as used in Fig. 1.9.
Figure 1.10. Generic temperature function with an asymptoted, eqn (1.41). Parameters are:
๐๐ = 20โ, ๐๐๐ = 5โ, ๐๐๐๐ก = 25โ, ๐ as indicated. Note that ๐(๐ = ๐๐) = 1.
1.3.6 Gompertz growth equation
Growth functions are widely applied in many branches of the biological sciences, generally in the analysis of
the time course of experimental data in plant and animal studies, although they can be adapted to respond
dynamically to changes in driving variables โ in the present model the Gompertz growth equation
(described in detail in this section) is used in the dynamic animal growth model in response to available
energy intake. The term growth function generally denotes an analytical function which can be written as a
single equation. Thus, a growth function for the time course of mass or weight ๐ is
W f t (1.42)
where ๐ก is time.
While a wide range of growth functions are used in agricultural modelling, there are two characteristics
which, as discussed by Thornley and Johnson (2000), they should satisfy if they are to be of significant use.
First, it should be derived from a differential equation for d๐ d๐กโ , so that there is an explicit equation for
the growth rate. Second, the parameters in the model should have some meaningful biophysical
interpretation so that variation in these parameters can be interpreted biologically. Thus,
,dW
g W tdt
(1.43)
which defines the growth rate in terms of the current weight and time. This choice of formulation implies
that the growth rate at any particular time also depends on the actual value of ๐. A further desirable
characteristic is that time can be eliminated from eqn (1.43) so that the growth rate can be written as a
โrate-stateโ equation โ that is, the growth rate is a function of the current state of the system which, in this
case, is ๐, so that eqn (1.43) can be written
dW
h Wdt
(1.44)
Once the growth rate has been defined by either eqn (1.43) or (1.44), the differential equation is solved for
๐(๐ก). However, if the equation is being applied with varying inputs, such as for animal growth, the
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0 10 20 30 40
Tem
per
atu
re fu
nct
ion
, f(T
)
Temperature, T (ยฐC)
q=1
q=2
q=3
DairyMod and the SGS Pasture Model documentation 14
equation must be solved numerically. (Numerical techniques are discussed in Section 1.4). Growth
functions are discussed in detail in Thornley and Johnson (2000) and Thornley and France (2007).
The Gompertz equation is derived from two equations. First, it is assumed that growth rate is directly
proportional to weight, so that
d
d
WW
t (1.45)
where ๐ d-1 is the specific growth rate and also is derived from a differential equation which is
d
dD
t (1.46)
where ๐ท is a parameter describing the decay of the specific growth rate. Integrating eqn (1.46) gives
0 e Dt (1.47)
with ๐0 being the initial value of ๐ (that is, the value at ๐ก = 0). Using eqn (1.47) in (1.45) gives
0
de
dDtW
Wt
(1.48)
which is the general formulation for the Gompertz differential equation.
To integrate eqn (1.48), it is written as
0
00
d
dW t Dt
W
We t
W (1.49)
where ๐(๐ก = 0) = ๐0 is the initial weight and the primes denote dummy variables of integration. This
equation is now easily solved to give
0
0
1
ln e DtW
W D (1.50)
and hence
00 1
exp e DtW W
D (1.51)
which is the Gompertz equation.
There are some useful characteristics of this equation. When ๐ก is small, using the series expansion of eโ๐ท๐ก,
1 e Dt Dt (1.52)
so that eqn (1.51) reduces to
0 0 expW W t (1.53)
This means that growth is approximately exponential during the early growth phase with specific growth
rate ๐. As ๐ก โ โ, ๐ approaches an asymptote given by
00
expfW W
D (1.54)
Differentiating eqn (1.51) gives
Chapter 1: Background to biophysical modelling 15
2
20
1
d d
e edd
Dt DtW WDW
tt (1.55)
Equating this to zero and substituting for d๐ d๐กโ from eqn (1.48), there is a point of inflexion at ๐กโ, where
01
lnt
D D and
e
fWW t t (1.56)
Thus, the point of inflexion occurs at about 37% of the final weight.
It was mentioned earlier that there are benefits in prescribing the growth rate equation independently of
time. This can be done by substituting for exp(โ๐ท๐ก) from (1.51) in (1.48) to give
00 0
1
dln
d
W D WW
t W (1.57)
In turn, either ๐ท or ๐0 can be eliminated using eqn (1.54) so that the Gompertz equation can be written as
00
lnd
d ln
f
f
W WWW
t W W (1.58)
or
d
lnd
fWWDW
t W
(1.59)
In practice, the parameter ๐0 is probably easier to prescribe from an intuitive understanding of the system
being modelled โ that is, the specific growth rate during early growth โ so that eqn (1.58) may be
preferable to eqn (1.59). Note that the maximum growth rate, which occurs at the inflexion point, can be
derived from eqns (1.56) and (1.58) as
0
0
d
d e lnmx
f
f
WW
t W W (1.60)
The Gompertz equation is illustrated in Fig. 1.11 using parameters that represent typical cattle growth,
where both the weight and growth rate are shown. The point of inflexion, which is the point where the
growth rate is maximum and as given by (1.56), is indicated. The equation has the typical sigmoidal growth
pattern that is generally observed for animals and that is also characteristic of plant growth. The Gompertz
equation is used in AgMod in the formulation of the animal growth model, but is adapted to allow for
fluctuating energy intake by the animals, as discussed in Chapter 6.
DairyMod and the SGS Pasture Model documentation 16
Figure 1.11. The Gompertz equation, eqn (1.51), illustrated with parameters that are typical
for cattle growth. The parameters are ๐0 = 40 kg, ๐๐ = 600 kg, ๐0 = 0.01 d-1, with ๐ท derived
from eqn (1.54). The red dotted lines indicate the point of inflexion, along with the weight and
growth rate at this point, as given by eqns (1.56) and (1.60).
1.4 Eulerโs method for solving differential equations
The dynamic models in the SGS Pasture Model and DairyMod are characterized by generally being written
as first order differential equations, that is, rate-state equations as discussed in the previous section for the
Gompertz equation. These equations generally cannot be solved analytically due to their complexity as
well as the fact that inputs are generally varying โ for example, climate for plant growth, available pasture
for animal growth, and so on. Consequently, the equations must be solved using numerical techniques.
Numerical methods for solving differential and other forms of equations is a large field in mathematics
known as numerical analysis. It is not my intention to discuss this in any great detail here, but an
introductory discussion is given in Thornley and Johnson (2000).
Most of the dynamic equations that have to be solved numerically in the present model have the form
d
,d
xf x t
t (1.61)
where ๐ฅ is the state variable, ๐ก is time, and ๐ denotes a function that defines the rate of change of ๐ฅ with ๐ก.
This is referred to as a rate-state equation. In practice, the function ๐ will also include model parameters
and environmental inputs, but these are not included here explicitly.
The simplest method for solving eqn (1.61) numerically is Eulerโs method. By definition,
0
dlim
d t
x t t x tx
t t (1.62)
Substituting eqn (1.61) into (1.62) gives
0
, limt
x t t x tf x t t
t (1.63)
For small values of ฮ๐ก it is approximately true that
,
x t t x tf x t t
t (1.64)
Rearranging this equation gives Eulerโs formula as
0
100
200
300
400
500
600
0 500 1000 1500
Emp
ty b
od
y w
eigh
t, k
g
Time, days
0.0
0.2
0.4
0.6
0.8
1.0
0 500 1000 1500
Gro
wth
rat
e, k
g d
-1
Time, days
Chapter 1: Background to biophysical modelling 17
,x t t x t t f x t t (1.65)
Thus, the state variable ๐ฅ at time ๐ก + ฮ๐ก is given as a function of its derivative at ๐ก and the time increment
ฮ๐ก. Equation (1.65) is also referred to as a forward difference equation.
As with any numerical technique, it is important to assess the accuracy of the method. This is not discussed
here, but the interested reader can read section 1.6 in Thornley and Johnson (2000). However, it can be
noted that for biophysical simulation models, Eulerโs method generally works well for pasture and animal
growth using a daily time-step, and is applied widely in the model.
1.5 Plant composition components
The theory presented here involves plant dry weight (d.wt) as well as photosynthetic rates. While the mole
is the preferred unit for photosynthesis, in most plant growth experiments the units of plant dry weight are
usually kg d.wt. To reconcile these units, it is assumed that the plant comprises sugars, which are mono
and disaccharides, protein, and cell wall material which is primarily cellulose and hemicelluloses. Other
components such as lipids are not considered here, although the analysis could be extended to include
them in a straightforward way. Taking the sugars to be primarily disaccharides (sucrose, fructose), the
following carbon compositions by mass are used:
1
1
1
0 44
0 53
0 42
Cell wall: . kg carbon kg cell wall
Protein: . kg carbon kg protein
Sugars: . kg carbon kg sugar
(1.66)
Denote the molar and dry weight fractions of plant material by
mole dry weight
Cell wall fraction: ๐๐ค ๐น๐ค
Protein fraction: ๐๐ ๐น๐
Sugar fraction: ๐๐ ๐น๐
It follows that the fraction of carbon in total plant dry weight is
0 44 0 53 0 42. . .C w p sF F F F (1.67)
The conversions between mole and dry weight fractions of the individual components are:
0 44
0 53
0 42
.
.
.
w w C
p p C
s s C
f F F
f F F
f F F
(1.68)
As an example, consider the plant dry weight composition to be 60% cell wall, 20% protein and 20% sugars
by weight, so that
๐น๐ค = 0.6, ๐น๐ = 0.2, ๐น๐ = 0.2, (1.69)
which gives (working to 2 significant figures)
๐น๐ถ = 0.45, (1.70)
DairyMod and the SGS Pasture Model documentation 18
and
๐๐ค = 0.58, ๐๐ = 0.23, ๐๐ = 0.09 (1.71)
Thus, while the mole and dry weight fractions of plant material do not vary greatly, they are, nevertheless,
not identical and appropriate care should be taken.
๐น๐ถ can be seen to be relatively insensitive to moderate changes in plant composition, although the carbon
content of the plant components will affect the calculation of ๐น๐ถ.
For whole plant material,
0 012.
1 mol C = kg d.wtCF
(1.72)
and, using eqn (1.70), this gives
(1.73)
or
371 kg d.wt mol C (1.74)
This is the conversion used here, although alternative values for ๐น๐ถ could readily be used. The parameter
๐พ = 37 mol C (kg d.wt)-1 (1.75)
will be used to convert from dry weight to mole units.
1.6 Atmospheric composition
Photosynthesis is influenced by atmospheric CO2 concentration, while transpiration and evaporation
depend on the water vapour concentration in the atmosphere. Methods for defining atmospheric gas
components are now considered.
Density is defined as kg m-3 and concentration as mol m-3. From the gas laws, the mole concentration of
any gas, ๐ (mol m-3), is given by
K
P
RT (1.76)
where ๐ (Pa) is the atmospheric pressure, ๐๐พ (K) is temperature and ๐ = 8.314 J K-1 mol-1 is the gas
constant. Note that, while ๐ถ is often used to define concentration in analysis of this type, ๐ is used here to
allow ๐ถ to be used in the treatment for CO2. Also, the notation ๐๐พ is used to avoid confusion with ๐ โ.
The atmosphere is taken to comprise the dry air components plus water vapour, with the principal
constituents of dry air (working to 2 percentage decimal places) being nitrogen (78.08%), oxygen (20.95%),
argon (0.93%), and carbon dioxide (0.04%). When water vapour is included, it can account for up to around
4% of the atmosphere (although this is subject to considerable variation) and in this case the proportions of
the main atmospheric constituents will decline slightly.
It is convenient to use either normal temperature and pressure (NTP) or standard temperature and
pressure (STP). NTP is usually taken to be 20ยฐC and 101.325 kPa. STP is 0ยฐC and 101.325 kPa. Note that
101.325 kPa is the SI definition of pressure and is equivalent to 1 atmosphere (atm) which, in turn, is
equivalent to 760 mm Hg and is a traditional value for atmospheric pressure at sea level. In all of the
analysis here, NTP will be used, since 20ยฐC is generally a more appropriate temperature for biological
processes than 0ยฐC, and is defined as:
NTP: 20ยฐC, 101.325 kPa. (1.77)
Chapter 1: Background to biophysical modelling 19
Thus, at NTP
41 574 .NTP mol m-3 (1.78)
which is the molar concentration of any gas.
The density ๐, kg m-3, is given by
M (1.79)
where ๐ (kg mol-1) is the molar mass, for example 44.01ร10-3 kg mol-1 for CO2.
In practice, the components of the atmosphere, such as CO2, O2, or water vapour are required. Denoting
the concentration of the atmosphere as ๐๐๐ก๐, eqn (1.76) can be rewritten as
atmK
P
RT (1.80)
The partial concentration of any component of the atmosphere, ๐๐ (mol m-3), such as CO2 or water vapour,
has concentration
ii
K
e
RT (1.81)
where ๐๐ (Pa) is the partial pressure of the gas.
The fractional concentration of gas ๐, ๐๐ mol gas ๐ (mol atmosphere)-1, is simply
i
iatm
c (1.82)
so that, using (1.80)
i iK
Pc
RT (1.83)
which defines the molar concentration in terms of the fractional concentration, atmospheric pressure, the
gas constant and temperature. As an example, consider CO2 at NTP, eqn (1.77), and with ๐๐ = 380 ๐mol
mol-1 (equivalent to 380 ppm), so that the true concentration of CO2 is
2
CO 0.01580 ๐mol m-3 (1.84)
Similarly, the partial pressure of constituent ๐, using (1.81) and (1.83), is
i ie c P (1.85)
so that the sum of the partial pressures of all the constituent gas components is equal to the atmospheric
pressure.
In general, fractional concentration is independent of temperature and pressure so that, for example, the
proportion of oxygen in the air at the top of Mount Everest is the same as at sea level, but the actual mole
concentration will decline. Thus, if ๐๐ is constant then eqn (1.83) implies that
iK
P
T (1.86)
and (1.85) that
ie P (1.87)
DairyMod and the SGS Pasture Model documentation 20
Although fractional molar concentration is generally used in models and analysis, the true mole
concentration, ๐๐, is arguably more appropriate for describing physiological processes as it defines the
absolute number of molecules per unit volume. As an example consider humans breathing oxygen. It is
common knowledge that we struggle at high altitudes. In this case, the fractional oxygen concentration is
the same as at sea level but the true concentration declines substantially. Clearly, our physiology is
responding to the true concentration. One possible reason why fractional concentrations are used in plant
physiology relates to the physiology of leaf photosynthesis, which is discussed in Chapter 3.
Now consider the air. Equation (1.79) gives
a a aM (1.88)
and for the constituent gases
i i iM (1.89)
For eqn (1.88) to be applied, it is necessary to derive an expression for the molar mass of air, ๐๐. This is
generally evaluated for dry air and the standard values for the molar mass and fractional concentration are
given in Table 1.1. Denoting the molar mass of dry air as ๐๐,๐๐๐ฆ, this is given by
2 2 2 2 2 2
,a dry N N O N Ar ar CO COM M c M c M c M c 0.02895 kg mol-1 (1.90)
If water vapour is present, as is usually the case, then ๐๐ will be slightly lower than ๐๐,๐๐๐ฆ, although the
difference is small (around 1%).
Table 1.1: Composition of dry air. ๐ is the molar mass and ๐ the fractional concentration. These values are taken from Monteith and Unsworth (2008), but adjusted so that
the CO2 fractional concentration is closer to current ambient.
Gas Nitrogen Oxygen Argon CO2
๐ (kg mol-1) 0.02801 0.03200 0.03898 0.04401
๐ (%) 78.08 20.95 0.93 0.04
1.6.1 CO2 concentration
As mentioned above, atmospheric CO2 concentration is often defined in parts per million, or ppm, which
refers to volume parts per million, and is equivalent to ๐mol mol-1, which is fractional molar concentration,
or mole fraction. Following convention, ๐ถ will be used to define CO2 concentration in units ยตmol CO2 mol
air, or ppm, and the current ambient CO2 is taken to be
๐ถ๐๐๐ = 380 ๐mol molโ1 (1.91)
Equation (1.83) can be applied to the fractional molar concentration of CO2, ๐ถ ยตmol mol, to give
2 610
COK
C P
RT (1.92)
so that, for example, at normal temperature and pressure, eqn (1.77)
2
0 01580 , .CO NTP ambC C mol CO2 m-3 (1.93)
and, taking the molar mass of CO2 to be 0.04401 kg mol-1 in eqn (1.89), the density is
2
3 32 20 0006953 0 6953 , . kg CO m . g CO mCO NTP ambC C (1.94)
Chapter 1: Background to biophysical modelling 21
1.6.2 Water vapour
Atmospheric water vapour content can be defined using the same approach as for CO2 above. However,
the two most common methods are to use vapour density, ๐๐ฃ kg H2O m-3, or vapour pressure, ๐๐ฃ Pa. Using
eqns (1.88) and (1.89) with (1.80) and (1.81) gives
v vv a
a
M e
M P (1.95)
where subscript ๐ฃ refers to water vapour. Assuming ๐๐ can be represented by ๐๐,๐๐๐ฆ, eqn (1.90), and
taking ๐๐ฃ = 0.01802, this becomes
0 622 . vv a
e
P (1.96)
It should be noted that in some texts the analysis leading to eqns (1.95) and (1.96) uses the density and
pressure for dry air and then combines that with the water vapour, rather than the present approach which
considers the total air composition including water vapour. This leads to a similar expression, but with the
term ๐ โ ๐๐ฃ in the denominator which is subsequently approximated to ๐ (see, for example, Thornley and
Johnson (2000) pp 423 and 633). With the present approach, it is necessary to assume that the molar mass
of air can be represented by the value for dry air. In practice, any errors are small and eqns (1.95) and
(1.96) can be used with confidence.
As the amount of water vapour in the air increases it eventually reaches saturation. The saturation vapour
pressure, ๐๐ฃโฒ , is related to temperature and is given by Tetens formula (Campbell and Norman, 1998)
17 5
611241
.expv
Te
T
Pa (1.97)
which defines ๐๐ฃโฒ in units of Pa, with ๐ in โ. The coefficients in (1.97) differ slightly from those given by
Allen et al. (1998), although the effect on ๐๐ฃโฒ is negligible. Equation (1.97) is illustrated in Fig. 1.12, with
units kPa rather than Pa.
Figure 1.12: Saturated vapour pressure, ๐๐ฃโฒ (kPa), as a function of temperature.
The vapour density, ๐๐ฃ, is often referred to as the absolute humidity, and the ratio of the actual vapour
density to saturated vapour density is the relative humidity, โ๐, that is
0
1
2
3
4
5
6
7
8
0 10 20 30 40
Satu
rate
d v
apo
ur
pre
ssu
re, k
Pa
Temperature, ยฐC
DairyMod and the SGS Pasture Model documentation 22
v v
rv v
eh
e (1.98)
where the prime denotes saturation and eqn (1.95) has been used to convert between pressure and
density. Relative humidity cannot exceed unity and is often expressed as a percentage.
Vapour pressure deficit is widely used and is the difference between the saturated and actual vapour
pressure, that is
v v ve e e (1.99)
which, using eqn (1.98), may be written
1 v v re e h (1.100)
Relative humidity is a simple unit to work with and has appeal. However, it has limitations in terms of
defining plant and canopy processes since for a given amount of water in the atmosphere it will vary
substantially in response to temperature. To illustrate this point, Fig. 1.13 (left) shows the relative humidity
as a function of temperature with ๐๐ฃ = 0.6๐๐ฃโฒ (๐ = 20โ) so that the relative humidity at 20ยฐC is 60%. It is
quite clear that the relative humidity will vary substantially for a fixed amount of atmospheric water
vapour. The corresponding vapour pressure deficit, in units kPa, is shown in Fig. 1.13 (right) which
demonstrates that the driving force for transpiration and evaporation, the vapour pressure deficit, will vary
in response to temperature for a fixed vapour pressure. Equation (1.100) should be used with caution and
(1.99) is preferable.
Figure 1.13: Left: relative humidity, โ๐ (%), as a function of temperature for vapour pressure
corresponding to 60% saturation at 20ยฐC. Thus, โ๐ = 60% at 20ยฐC in this illustration.
Right: the corresponding vapour pressure deficit.
1.7 Final comments
The theory described in this Chapter covers the mathematical concepts of the background topics that are
required for the various modules in the SGS Pasture Model and DairyMod. It is intended that this Chapter
provides all of the necessary background for the full model description in the remainder of this book.
However, many of the topics presented in this Chapter are also discussed in Thornley and Johnson (2000)
and Thornley and France (2007). These texts also cover a wide range of models and modelling approaches
in plant and crop physiology, and agricultural simulation modelling in general.
0
20
40
60
80
100
120
0 10 20 30 40
Re
lati
ve h
um
idit
y, %
Temperature, ยฐC
0
1
2
3
4
5
6
7
0 10 20 30 40Vap
ou
r p
ress
ure
def
icit
, kP
a
Temperature, ยฐC
Chapter 1: Background to biophysical modelling 23
1.8 References
Allen RG, Pereira LS, Raes D and Smith M (1998). FAO irrigation and drainage paper no. 56: crop
evapotranspiration. www.kimberly.uidaho.edu/ref-et/fao56.pdf.
Ball JT, Woodrow IE & Berry JA (1987). A model predicting stomatal conductance and its contribution to
the control of photosynthesis under different environmental conditions. In: Progress in Photosynthesis
Research, vol IV, ed J Biggens. Martinus Nijhoff / American Society of Agronomy, Dordrecht, the
Netherlands / Madison, WI, USA, 221-224.
Blonquist Jr JM, Norman JM & Bugbee B (2009). Automated measurement of canopy stomatal
conductance based on infrared temperature. Agricultural and Forest Meteorology,149, 1931-1945.
Bunce JA (2000). Responses of stomatal conductance to light, humidity and temperature in winter wheat
and barley grown at three concentrations of carbon dioxide in the field. Global Change Biology, 6, 371-
382.
Campbell GS and Norman JM (1998). An Introduction to environmental biophysics, second edition.
Springer, New York, USA.
Gale, J (2004). Plants and altitude โ revisited. Annals of Botany, 92(4), 199.
Johnson and Thornley (1985). Temperature dependence of plant and crop processes. Annals of Botany,
55, 1-24.
Jones HG (1992). Plants and microclimate. Cambridge University Press, Cambridge, UK.
Mackowiak CL, Wheeler RM & Yorio NC (1992). Increased leaf stomatal conductance at very high carbon
dioxide concentrations. HortScience, 27, 683-684.
Monteith JL & Unsworth MH (2008). Principles of environmental physics, third edition. Elsevier, Oxford,
UK.
Thornley JHM (1998). Grassland dynamics: an ecosystem simulation model. CAB International,
Wallingford, UK.
Thornley JHM and France J (2007). Mathematical models in agriculture. CAB International, Wallingford, UK.
Thornley JHM & Johnson IR (2000). Plant and crop modelling. Blackburn Press, Caldwell, New Jersey, USA.
DairyMod and the SGS Pasture Model documentation 24
2 Climate
2.1 Introduction
Plant growth responds directly to climatic conditions, in particular light (irradiance), temperature, rainfall,
atmospheric CO2 concentration, vapour pressure deficit and windspeed. The model underlying the SGS
Pasture Model and DairyMod has been developed to use SILO climate data files (Jeffrey, 2001), which is a
tremendous resource available for pasture and crop modelling in Australia. SILO files provide daily data for
locations on a 0.05โฐ longitude and latitude grid across Australia, which is a grid of approximately 5.5km,
from the late 1800s until yesterday. Daily data that are used in AgMod are rainfall, maximum and
minimum temperature, total solar (or global) radiation, and vapour pressure. Windspeed data are not
routinely available and so it is necessary to assume a fixed value for windspeed in the model. Atmospheric
CO2 is prescribed by the user and it is quite straightforward to work with variable CO2 concentration. SILO
also provides an estimate of the FAO56 (Allen et al., 1998) estimate of reference evapotranspiration and
this is used in the model interface to analyse long-term trends in rainfall surplus, that is rainfall in excess of
potential evapotranspiration demand, for the location being studied. This reference evapotranspiration is
also used in the irrigation strategy based on rainfall deficit (which is just the negative value of rainfall
surplus), but it is not used in any of the evapotranspiration calculations in the model.
The key biophysical responses to climate are photosynthesis and transpiration. Photosynthesis responds to
the photosynthetically active component of solar radiation, as discussed in Chapter 1, as well as
temperature and CO2 concentration. Transpiration is dependent on the longwave radiation balance of the
canopy, and also temperature and vapour deficit, as well as leaf stomatal conductance and the canopy
architecture through their influence on the transfer of water vapour from the canopy to the bulk air
stream. Atmospheric CO2 concentration and vapour pressure deficit were discussed in Chapter 1. The
climate inputs required for the model are discussed in turn.
2.2 Rainfall
Rainfall inputs are applied from above the canopy so that they are first intercepted by the canopy and then
the litter and ground, as illustrated in Fig. 2.1.
Rainfall intensity can be important in water dynamics, in particular with relation to runoff. Since the model
is designed to work with daily rainfall data, it is necessary to estimate rainfall intensity. This is done by
allowing the user to specify the daily rainfall duration in hours, either as a constant for all months, or for
individual months of the year. Thus, for example, it is possible to characterise regions that may have a high
incidence of summer storms but where winter rainfall is generally light. Using the daily rainfall duration,
the model then randomly selects a start time for the rain for each day of the simulation. This uses the in-
built random number generator in the development programming language, Delphi (Embarcadero),
although it should be noted that the random number generator is โseededโ to ensure the same sequence of
random numbers is generated each time a specific climate file is loaded.
Chapter 2: Climate 25
Figure 2.1: Schematic representation of rainfall inputs
2.3 Temperature
The model is designed to work with maximum and minimum daily temperature data, ๐๐๐ฅ and ๐๐๐
respectively, with the mean daily temperature, ๐๐๐๐๐, defined simply as the average of these values:
0 5 .mean mx mnT T T (2.1)
For some processes, such as plant maintenance respiration, it is sufficient to work with ๐๐๐๐๐, although for
others the day or night temperature may be required, for example in the calculations for day and night soil
evaporation (although most evaporation will generally occur during the day). Representative day and night
temperatures are taken to be
2 3 4
2 3 4
day mx mean mx mn
night mn mean mx mn
T T T T T
T T T T T (2.2)
2.4 Radiation
Radiation plays a crucial role in plant, crop and pasture physiological processes. The visible component of
solar, or shortwave, radiation is the fundamental energy source for photosynthesis. The longwave
radiation that is emitted by terrestrial bodies as well as the atmosphere, combined with solar radiation, is
the key energy driver for the evaporation of water. A background to the basic principles relating to
radiation components is now presented, but for a more complete discussion see, for example, Jones (1992),
Campbell and Norman (1998), Monteith and Unsworth (2008).
A background to the basic physics of radiation is presented, followed by the analysis required for canopy
photosynthesis and energy balance, which includes the calculations of evaporation and transpiration in
response to environmental conditions. The radiation analysis forms the bulk of this Chapter. Some of the
symbols in the early sections are used with different definitions later for canopy calculations.
canopy
litter
soil
Rain
DairyMod and the SGS Pasture Model documentation 26
The models presented in this Chapter can be explored in detail in the software package PlantMod (Johnson,
2013), and some of the illustrations presented here are taken directly from that program.
2.4.1 Black body radiation
The energy level distribution from a body is a function of its temperature, its wavelength and surface
properties. If the surface properties are such that there is no reduction in energy emitted due to those
surface properties, then that body is referred to as a black body. Planckโs law, published in 1900, is derived
from quantum mechanics, and states that
2
5
2 1
1
,
KK hc kT
hcI T
e (2.3)
where ๐ผ(๐, ๐๐พ) is the spectral emittance, W m-2 (m wavelength)-1 (sometimes written as W m-3), which is
the energy per unit surface area per unit wavelength of the emitting body as a function of its temperature
๐๐พ (K) (the subscript ๐พ is used to differentiate from ยฐC), and wavelength ๐ (m) of the emitted radiation.
โ = 6.626 ร 10โ34 J s is Planckโs constant; ๐ = 2.998 ร 108 m s-1 is the speed of light; and ๐ = 1.3807 ร
10โ23 J K-1 is the Boltzmann constant. ๐ผ(๐, ๐) is illustrated in Fig. 2.2 for ๐๐พ = 6000 K, which approximates
to the external surface of the sun, and ๐๐พ = 300 K (equivalent to 27ยฐC) which is representative of
temperatures on earth. The scales of these graphs are different since the energy emitted by the sun is
much greater than that from the surface of the earth. The ultraviolet, photosynthetically active (see below)
and infrared components of solar radiation are also indicated. It can be seen that ๐ผ(๐, ๐๐พ) peaks at a
shorter wavelength for the higher temperature, and Weinโs law can be derived which states that the peak
wavelength is
6289710 m
KTm (2.4)
so that ๐๐(๐๐พ = 6000) = 0.480 ยตm and ๐๐(๐๐พ = 300) = 9.597 ยตm, which means that the peak energy
emitted by terrestrial bodies has a much longer wavelength than for the sun โ hence the terms shortwave
and longwave. Longwave radiation is sometimes referred to as terrestrial or far-infrared radiation.
The total energy emitted at a particular temperature is the integral of ๐ผ(๐, ๐๐พ) for all wavelengths, which is
the area under the curve shown in Fig. 2.2. This can be derived from eqn (2.3) and is known as the Stefan-
Boltzmann equation, which (for a black body) is
4 KE T (2.5)
with units W m-2, where ๐ = 5.670 ร 10โ8 W m-2 K-4 is the Stefan-Boltzmann constant. Thus, ๐ธ(๐๐พ =
6000) = 7.35 ร 107 W m-2 โก 73.5 MW m-2 and ๐ธ(๐๐พ = 300) = 459 W m-2, demonstrating the greater
energy emitted per unit area of the sun compared with the earth.
Of the radiation emitted by the sun about half of it is visible and this is the middle part of the frequency
distribution, ranging from around 0.4 ยตm (blue light) to 0.7 ยตm (red light). Ultraviolet (UV) radiation is the
component from below 0.4 ยตm, while the infrared (IR) component is the range 0.7 to 3 ยตm. These values,
which are indicated in Fig. 2.2, are not exact since there is no sharp transition between the different
components. The visible component of radiation is also known as photosynthetically active radiation (PAR)
as this is the component of radiation that provides the energy for photosynthesis โ this is discussed below.
The wavelength of the radiation emitted by terrestrial bodies covers the range from about 3 to 100 ยตm,
which is much greater than that for the sun, and hence is referred to as longwave, or far-IR, radiation.
Chapter 2: Climate 27
Figure 2.2: Spectral distribution of radiation emitted from black bodies for ๐๐พ = 6000 K (left)
and ๐๐พ = 300 K (right), which correspond roughly to the surface of the sun and earth
respectively, as a function of wavelength. The red line is shortwave, or solar, radiation, and
the green line is longwave, or terrestrial, radiation. Also indicated for the shortwave radiation
are the ultraviolet (UV), photosynthetically active radiation (PAR) and near-infrared (IR). The
longwave radiation is also known as far-infrared. The components of radiation are discussed in
the text. Note the scale for the shortwave radiation axis (left) is 106 times that on the right,
which highlights the much greater energy emission by the sun.
2.4.2 Non-black body and grey body radiation
In practice, most terrestrial bodies do not behave like perfect black bodies and eqn (2.5) is modified to give
4K
E T (2.6)
where ํ, 0 < ํ โค 1, (dimensionless) is the emissivity. Equation (2.6) applies to bodies where the emissivity
is independent of wavelength. Black bodies therefore have an emissivity of 1 and bodies with ํ < 1 are
referred to as grey bodies. For most natural surfaces (including snow) ํ lies between 0.95 and 1. Although
it is reasonable to use the value 1, it is assumed that
0 97 . (2.7)
for calculation in the model.
2.4.3 Radiation energy for photosynthesis: PAR and PPF
As mentioned earlier, the visible component of the radiation emitted by the sun, which is in the range 0.4
to 0.7 ยตm (or 400 to 700 nm), provides the energy for photosynthesis. This is referred to as
photosynthetically active radiation, or PAR, and is commonly assumed to be around half the total solar
radiation. However, the precise fraction depends on climatic factors such as cloud cover and solar
elevation. From the Clear Sky Calculator (www.clearskycalculator.com), it can be seen that a more accurate
conversion is for PAR to be 45% of the total solar radiation. This fraction is not fixed, but increases when
humidity increases and can reach close to 50%. The units for describing PAR are W m-2 โก J m-2 s-1.
0
20
40
60
80
100
120
0 0.5 1 1.5 2
Spec
tral
em
itta
nce
at
60
00
K, M
W m
-2 ยต
m-1
Wavelength, ยตm
PAR UV near IR
0
5
10
15
20
25
30
35
40
0 20 40
Spec
tral
em
itta
nce
at
30
0K
, W m
-2 ยต
m-1
Wavelength, ยตm
DairyMod and the SGS Pasture Model documentation 28
For photosynthesis studies, the energy is generally expressed as the molar flux of photons between 0.4 to
0.7 ยตm, and is referred to as photosynthetic photon flux, or PPF. The term PPF will be used throughout this
discussion for the definition of the energy source for photosynthesis.
There is no precise conversion between PAR and PPF that can be applied for all atmospheric conditions
since, as discussed above, the energy of the radiation depends on the wavelength and so depends on the
spectral composition of the light. A reasonable value, using the Clear Sky Calculator and based on summer
conditions at Logan, Ut, (Bruce Bugbee, pers. comm., www.clearskycalculator.com) is
1 ยตmol photons PAR โ 0.218 J PAR. (2.8)
Finally, note that the term photon flux density, or PFD, is sometimes used, but this is now discouraged in
the literature. Iโm grateful to Professor Bruce Bugbee for clarifying these definitions.
2.4.4 Radiation units and terminology
In the theory presented above, both J m-2 s-1 (equivalent to W m-2) and ๐mol photons m-2 s-1 have been
used for radiation energy. In the physics literature, J m-2 s-1 (W m-2) is almost universally used, while in
photosynthesis studies the recent trend has been towards ๐mol photons m-2 s-1. In the present theory,
both sets of units will be used, although this should not cause confusion. For discussions of photosynthesis,
the convention for using ๐mol photons m-2 s-1 (PPF) is followed, but for energy dynamics and evaporation, J
m-2 s-1 units are preferred, and this is referred to as irradiance which is the total solar radiation and not just
the photosynthetically active component. The choice of J m-2 s-1 rather than W m-2, which are equivalent, is
because daily radiation values are also used which have units J m-2 d-1.
2.4.5 Canopy light interception and attenuation
In order to model canopy photosynthesis in response to environmental factors, it is necessary to develop
models of light interception and attenuation through the depth of the canopy. The canopy photosynthetic
rate is then calculated in terms of the photosynthetic response to PPF of the leaves within the canopy and
the variation of PPF through the depth of the canopy. The approach taken here is to look at the mean PPF
through the canopy as well as the components of direct and diffuse sunlight. In doing so, the PPF
components within the canopy and those that are actually incident on the leaf surfaces are considered.
This component of the theory deals with the PPF with units ๐mol photons m-2, where the area unit can
refer either to ground or the leaf. The model for light interception and attenuation can be explored in
PlantMod and so only a few illustrations are presented here.
Mean PPF
First consider the mean PPF within the canopy. In overcast conditions, there will be little variation in PPF in
the horizontal plane and so this approach is applicable without modification. However, for clear skies with
strong sunflecks in the canopy, there will be considerable horizontal variation in the PPF. The following
theory still applies to these situations but it has to be extended to identify the direct and diffuse
components of PPF.
As light is intercepted and absorbed by leaves within the canopy, the PPF declines, as described by Beer's
law:
0 e kI I (2.9)
where ๐ผ0 is the PPF incident on the canopy and ๐ is the canopy extinction coefficient. A derivation of this
equation is given in Thornley and Johnson (2000, Chapter 8). Note that since โ has dimensions (m2leaf) (m-2
ground) it follows that ๐ has dimensions (m2 ground) (m-2 leaf), and hence eqn (2.2) is dimensionally
Chapter 2: Climate 29
consistent. However, for most purposes it is sufficient to regard ๐ as dimensionless. The PPF through the
canopy as described by eqn (2.9) is illustrated in Fig. 2.3 for ๐ = 0.5 which is typical of cereals and grasses,
and ๐ = 0.8 which is appropriate for canopies with more horizontally inclined leaves.
Figure 2.3: Mean PPF as a function of cumulative leaf area index through the canopy, as given
by eqn (2.9) for ๐ = 0.5 (solid) and ๐ = 0.8 (dash), and with the PPF incident on the canopy
given by ๐ผ0 = 1000 ยตmol m-2 s-1. From PlantMod.
A simple interpretation of the extinction coefficient, ๐, is the cosine of the angle between the leaves and
the horizontal plane. Thus, for perfectly horizontal leaves ๐ = cos(0) = 1.
Equation (2.9) defines the PPF per unit horizontal (or ground) area, but the PPF per unit leaf area is
required in order to calculate the rate of photosynthesis of the leaves in the canopy. This is given by
I kI (2.10)
where the factor ๐ projects the leaf area index onto the horizontal plane โ for a derivation, see Thornley
and Johnson (2000, p. 203)
Direct and diffuse PPF
The theory is now developed to identify the direct and diffuse components of PPF within the canopy. This
approach is based on the early work of Norman (1980, 1982) as well as that by Campbell (1977) and Stockle
and Campbell (1985). This treatment of direct and diffuse PPF is widely used and the analysis presented
here closely follows Johnson et al. (1995) and has been applied, for example, by Thornley (2002).
Using subscripts ๐ and ๐ to denote the direct solar beam and diffuse PPF respectively, above the canopy,
the PPF is
0 0 0 , ,s dI I I (2.11)
Within the canopy at leaf area index โ it is
s dI I I (2.12)
and the corresponding PPF incident on the leaf surfaces is
, ,s dI I I (2.13)
It is assumed that the direct and diffuse components of ๐ผ0 decline through the depth of the canopy
according to eqn (2.9). Defining ๐๐ as the fraction of total radiation that is direct, so that
0.0 1.0 2.0 3.0 4.0 5.0
Leaf area index
0
200
400
600
800
PP
F, ยต
mo
l / m
2.s
DairyMod and the SGS Pasture Model documentation 30
0 0,s sI f I (2.14)
it therefore follows that
0 e k
s sI f I (2.15)
and
01 e kd sI f I (2.16)
To calculate the incident PPF on the leaves, it is necessary to evaluate the components of leaf area index
which are in direct sunlight and diffuse PPF, denoted by โ๐ and โ๐ respectively. โ๐ is obtained by noting
that the reduction in the direct beam is intercepted by โ๐ , so that
0 0 1 , , e ks s sk I I (2.17)
from which
1
e k
sk
(2.18)
The factor ๐ on the left hand side of eqn (2.17) is required as this projects the leaf area index, โ๐ , onto the
horizontal plane. The remainder of the leaves are in diffuse PPF and have leaf area index
d s (2.19)
The incident PPF on โ๐ is, applying eqns (2.10) and (2.16)
0 1
,
e
d d
ks
I kI
kI f (2.20)
The PPF incident on โ๐ is the combination of the diffuse component and the direct solar beam, as given by
0
0 1
, , ,
e
s s d
ks s
I kI I
kI f f (2.21)
Note that Norman (1982) relates the extinction coefficient, k to solar elevation. While this is a good
objective for detailed study of light interception in canopies, difficulties arise when looking at mean values
over the day (for further discussion, see Johnson et al., 1995). One value for the extinction coefficient for
both direct and diffuse PPF is used and it is assumed to be constant.
The mean PPF, ๐ผ, direct and diffuse components, ๐ผ๐ , ๐ผ๐, and PPF incident on leaves in direct and diffuse
light, ๐ผ๐,๐ , ๐ผ๐,๐, are shown in Fig. 2.4.
The theory presented here provides a complete description of the attenuation and interception of the
direct and diffuse PPF components through the canopy. It is necessary to define the canopy light extinction
coefficient, ๐, the PPF on the canopy, ๐ผ0, and the fraction of ๐ผ0 that is from the direct solar beam, ๐๐ . In
practice, the extinction coefficient may vary from around 0.5 for cereals and grasses to 0.8 for species with
more horizontally inclined leaves such as clover. For skies where the sun is not obscured by cloud, ๐๐ may
be typically around 0.7, although this can depend on atmospheric composition.
Chapter 2: Climate 31
Figure 2.4: The mean PPF, ๐ผ, direct and diffuse components, ๐ผ๐ , ๐ผ๐, and PPF incident
on leaves in direct and diffuse light, ๐ผ๐,๐ , ๐ผ๐,๐, with ๐ = 0.5. From PlantMod.
There are some simplifying assumptions in this approach, as discussed by Thornley and France (2007). The
main simplifications are that the leaves are assumed to be randomly distributed (see Thornley and Johnson,
2000, for a discussion on leaf distribution), and variation in the direction of the direct solar beam
throughout the day is not included. Also, light reflection and transmission through the leaves has not been
incorporated directly, although in photosynthesis studies, the rate of leaf photosynthesis is generally
calculated in terms of incident light and not absorbed light. By working with eqn (2.9) directly (Beerโs law),
it is reasonable to assume that the reflected and transmitted components are incorporated implicitly.
While the analysis is relatively simple, this level of complexity is widely used in crop and pasture studies for
the calculation of canopy photosynthesis, which is considered in Chapter 3.
Ground cover
In the analysis for the canopy energy balance the fractional ground cover is required. According to eqn
(2.9), the solar radiation that is transmitted through the canopy is
0 kL
tI I e (2.22)
Defining the fractional ground cover, ๐๐, as the proportion of solar radiation that is not transmitted, it
follows that
1 kLgf e (2.23)
This simple expression will be used in the analysis for the canopy radiation balance.
2.4.6 Clear-sky solar radiation and daylength
In the treatment of canopy transpiration, temperature and energy balance, it is necessary to estimate the
clear-sky daily solar radiation, ๐ ๐,0, MJ m-2 day. The theory presented here follows Campbell (1977) and
Thornley and France (2007).
Three standard equations relating to the geometry of the earthโs rotation and its orbit around the sun are
first presented without derivation. These are the solar declination angle, ๐ฟ (rad), which is the angle
between the earthโs equatorial plane and the line from the earth to the sun, and accounts for the tilt of the
earthโs axis relative to the sun; the solar elevation angle at local noon, ๐ (rad); and the daylength, ๐๐, as a
fraction of the 24 hour period. If ๐ก is the day of year from 1 January, ๐ (rad) the latitude, then
0.0 1.0 2.0 3.0 4.0 5.0
Leaf area index
0
200
400
600
800
PP
F, ยต
mo
l / m
2.s
Imean
Is
Id
Ils
Ild
DairyMod and the SGS Pasture Model documentation 32
81
23 45 2180 365
. sin
t (2.24)
1 sin sin sin cos cos (2.25)
11cos tan tandayf
(2.26)
and the number of daylight hours per day is
24 day dayh f (2.27)
If ๐ is prescribed in degrees then, using obvious notation,
180
deg (2.28)
To convert ๐ฟ and ๐ to degrees, multiply by 180 ๐โ . Also note that latitudes in the northern hemisphere are
positive while they are negative in the southern hemisphere.
๐ฟ, ๐ and โ๐๐๐ฆ are illustrated in Fig. 2.5 for elevations of 0 (the equator), 20, 40, 60ยฐ. It can be seen that the
solar declination is positive in summer, negative in winter and zero at the spring and autumn equinoxes
(which are around 21 March and 22 Sept). For latitudes outside locations between the tropic of Cancer and
Capricorn, which are ยฑ 23.4ยฐ respectively, the solar elevation angle at local noon, ๐, is maximum in the
middle of summer. However, at the equator, ๐ is maximum at the spring and autumn equinoxes and,
moving towards the tropics, the two maxima converge. Finally, the daylength follows a familiar pattern and
is seen to be fixed at 12 hours for the equator while, for other locations it is, of course, greater in summer,
with longer days and shorter nights as the latitude increases.
Turning to irradiance, three sets of variables are used with appropriate subscripts (๐ ๐ has already been
defined above). These are:
๐ฝ, instantaneous irradiance outside the earthโs atmosphere: W m-2 or J m-2 s-1;
๐ผ, instantaneous irradiance at the earthโs surface: W m-2 or J m-2 s-1;
In all cases, irradiance is measured parallel to the horizontal plane at the surface of the earth.
The irradiance outside the earthโs surface at solar noon, ๐ฝ๐๐๐๐, is
sinnoonJ (2.29)
where
1367 W m-2 s-1 (2.30)
is the solar constant, and is the irradiance perpendicular to the sun at the edge of the earthโs atmosphere.
It is now assumed that the potential, or clear-sky, irradiance at the earthโs surface, ๐ผ๐,๐๐๐๐, is given by
,p noon noonI J (2.31)
where ๐ is an atmospheric diffusivity coefficient. While more complex equations have been used this
approach works well for a range of locations in Australia, as will be seen shortly, and there is no obvious
reason to suggest other locations will behave much differently. Comparisons with experimental data
suggest
0 73 . (2.32)
Chapter 2: Climate 33
is a good default value, although it may be necessary to adjust this parameter for different sites. However,
this is relatively easy to estimate, as discussed below. It should be noted that a slightly different approach
than a fixed constant in eqn (2.31) is used by Allen et al, (1998) although, as will be seen below, the
present approach works well. ๐ผ๐,๐๐๐๐ is illustrated in Fig. 2.6 for the latitudes used in Fig. 2.5. It can be
seen that the variation in ๐ผ๐,๐๐๐๐ is most apparent as the latitude moves further from the equator. It peaks
at the equinoxes at the equator and, outside the tropics, it peaks in mid-summer, with the range between
summer and winter increasing as the distance from the equator increases.
Figure 2.5: Solar declination angle, ๐ฟ (ยฐ), solar elevation angle at local noon, ๐ (ยฐ),
and the daylength, โ๐๐๐ฆ (hours), as given by eqns (2.24) to (2.27). The latitude is as indicated
(๐ฟ is independent of latitude). Note that in the theory ๐ฟ and ๐ are
prescribed in radians.
-30
-20
-10
0
10
20
30
1/01 1/02 1/03 1/04 1/05 1/06 1/07 1/08 1/09 1/10 1/11 1/12 1/01
Sola
r d
eclin
atio
n (
ยฐ)
0
10
20
30
40
50
60
70
80
90
100
1/01 1/02 1/03 1/04 1/05 1/06 1/07 1/08 1/09 1/10 1/11 1/12 1/01
Sola
r el
evat
ion
at
no
on
(ยฐ)
0ยฐ
20ยฐ
40ยฐ
60ยฐ
0
2
4
6
8
10
12
14
16
18
20
1/01 1/02 1/03 1/04 1/05 1/06 1/07 1/08 1/09 1/10 1/11 1/12 1/01
Day
len
gth
(h
ou
rs)
0ยฐ
20ยฐ
40ยฐ
60ยฐ
DairyMod and the SGS Pasture Model documentation 34
Figure 2.6: Noon potential solar radiation, ๐ผ๐,๐๐๐๐, as given by eqn (2.31).
To calculate the potential daily solar radiation, it is assumed that the potential solar radiation throughout
the day, ๐ผ๐, varies sinusoidally, and can be written
, sin ,p p noonI I d d 0โ1 over the daylight period (2.33)
so that the mean daily potential, or clear-sky, irradiance is
2
86 400
, ,,S p day p noonR f I (2.34)
where 86,400 is the number of seconds in a day. Thus,
2
86 400
, , sinS p dayR f (2.35)
This is a simple equation for the clear-sky irradiance in terms of latitude and day of year, and is illustrated in
Fig. 2.7 for the latitudes used in Figs 2.5 and 2.6. The general trend for ๐ ๐,๐ is similar to that for ๐ผ๐,๐๐๐๐,
although the maximum value for ๐ ๐,๐ is greater as latitudes increase, whereas this is not the case for
๐ผ๐,๐๐๐๐. This difference is due to the fact that the maximum daylength increases at higher latitudes and the
total potential daily solar radiation (๐ ๐,๐) is the combination of the potential instantaneous solar radiation
(๐ผ๐) and daylength.
0
200
400
600
800
1000
1200
1/01 1/02 1/03 1/04 1/05 1/06 1/07 1/08 1/09 1/10 1/11 1/12 1/01
No
on
po
ten
tial
so
lar
rad
(J
m-2
s-1
) 0ยฐ
20ยฐ
40ยฐ
60ยฐ
Chapter 2: Climate 35
Figure 2.7: Clear-sky potential solar radiation, ๐ ๐,๐ (MJ m-2 d-1), as given by eqn (2.35) for the
latitudes as indicated.
In order to test the approach, data from two sites in Australia from 1901 to 2008 are used from the SILO
data set (Jeffrey, 2001). These sites are Barraba, NSW, at latitude -30.5ยฐ, and Albany, WA, at latitude -35ยฐ.
Potential irradiance for each day of the year is estimated as the maximum observed for each day in the
climate file. This assumes, therefore, that for each day of the year there was at least one occasion in the
108 years where the sky was clear. Figure 2.8 shows the observed and predicted clear-sky irradiance. Also
shown are the mean and minimum irradiance, to illustrate the range of values that occur. There are
occasional โblipsโ in the data for maximum and minimum irradiance, but these may well be due to
fluctuations in the accuracy of the measurement equipment. The data and model for clear-sky irradiance
are virtually identical which gives confidence in the theoretical approach.
Figure 2.8: Observed maximum daily irradiance (blue) and predicted values (red) using eqn
(2.35). Note that the blue lines are obscured for much of the data due to the close similarity of
the values. Also shown are the mean (green) and minimum (purple) daily irradiance values.
Equation (2.35) has been tested for several other sites around Australia with similar close agreement with
the data. The key parameter defining potential solar radiation is the atmospheric diffusivity, ๐, which may
vary for different locations, although the value 0.73 has been found to be appropriate in many cases. This
equation will be used in Chapter 4 which considers canopy transpiration, temperature and energy budget.
0
5
10
15
20
25
30
35
40
1/01 1/02 1/03 1/04 1/05 1/06 1/07 1/08 1/09 1/10 1/11 1/12 1/01
Cle
ar s
ky s
ola
r ra
dia
tio
n, M
J m
-2 d
-1
0ยฐ
20ยฐ
40ยฐ
60ยฐ
0
5
10
15
20
25
30
35
1/01 1/03 1/05 1/07 1/09 1/11 1/01
MJ
m-2
d-1
Barraba
0
5
10
15
20
25
30
35
1/01 1/03 1/05 1/07 1/09 1/11 1/01
MJ
m-2
d-1
Albany
DairyMod and the SGS Pasture Model documentation 36
2.4.7 Net radiation
The net radiation balance, which accounts for both shortwave and longwave radiation components, is
central to the treatment of canopy transpiration, temperature and energy budget. Radiation is generally
described for instantaneous (s) or daily (d) time scales and detailed discussions of the underlying physics
relevant to plant canopies can be found, for example, in Thornley and France (2007) and Johnson (2013).
The present discussion describes the net radiation balance of a canopy as required for the description of
canopy transpiration.
The radiation balance of the canopy is shown in Fig. 2.. Solar radiation is incident on the canopy, which is
either reflected and absorbed by the canopy or transmitted through the canopy. Longwave radiation
transmitted from the atmosphere is intercepted and absorbed by the canopy with a small component that
is reflected (and often ignored). Longwave radiation is also emitted by the canopy as a function of its
temperature. Both shortwave and longwave radiation components that are not intercepted by the canopy
will be transmitted through the canopy.
The theory for the radiation balance of canopies is presented with the assumption of full ground cover. In
Chapter 4 canopy transpiration is calculated for ground cover and then adjusted for partial cover. The
theory with partial ground cover is discussed by Johnson (2013).
Figure 2.9: Schematic representation of the radiation balance of the canopy.
Denoting the incoming shortwave radiation by ๐ ๐ J m-2 d-1, the absorbed solar radiation as ๐ ๐,๐ J m-2 d-1is
1,S a SR R (2.36)
where the reflection coefficient, or albedo, ๐ผ, has default value
0 23. (2.37)
Longwave, or terrestrial radiation, is the radiation emitted by a body as a function of its temperature.
Incoming longwave radiation from the atmosphere depends primarily on atmospheric properties and
temperature, generally increasing in response to vapour density and cloud cover. Note that in the analysis
presented here, the net longwave radiation term is defined as outgoing rather than incoming. In some
Incoming solar
Reflected solar
Emitted longwave
Incoming longwave
Canopy
Transmitted solar and longwave
Absorbed radiation
Soil heat flux
Reflected longwave
Chapter 2: Climate 37
texts, it is defined as incoming to be consistent with incoming solar radiation. The choice to use outgoing
here is because this term is generally positive.
The present approach uses the empirical equation of Allen et al. (1998, eqn (39)) to define the daily net
outgoing longwave radiation as
486 400 0 34 0 14 1000 1 35 0 35, , ,,
, . . . .SL n K day v a
S p
RR T e
R
(2.38)
where ๐๐พ (K) is the mean daily temperature in Kelvin units, ๐ is the Stefan-Boltzmann constant, where
๐ = 5.6704 ร 10โ8 J m-2 s-1K-4, ๐๐ฃ,๐ (Pa) is vapour pressure with the factor 1000 converting this to kPa, ๐ ๐
(J m2 d-1) as defined above is daily solar radiation, and ๐ ๐,๐ (J m2 d-1) is the potential, or clear-sky, solar
radiation with ๐ ๐ โค ๐ ๐,๐. Clear-sky solar radiation was derived in terms of the latitude and day of year in
section 2.6 above. The factor 86,400 is the number of seconds per day and converts the radiation
components in the Stefan-Boltzmann equation to daily values. According to Allen et al. (1998), the
(0.34 โ 0.14โ๐๐ฃ,๐ 1000โ ) term corrects for air humidity and declines as humidity increases. The
(1.35 ๐ ๐ ๐ ๐,๐โ โ 0.35) term incorporates the influence of cloud cover and decreases as cloud cover
increases, since this will result in a reduction in ๐ ๐. The approach of eqn (2.38) is widely used โ for example
the SILO dataset (Jeffrey, 2001) available in Australia, which gives access to daily climate data from the late
1800s to the present day for any location in Australia, uses this approach for the calculation of the net
radiation balance. Note that Allen et al. (1998) incorporated the daily maximum and minimum
temperatures, although using a single mean daily temperature gives very similar results.
Combining eqns (2.36) and (2.38), the daily net radiation during the daylight period is
, , ,n day s a day L nR R f R (2.39)
where ๐๐๐๐ฆ is the daylight fraction and ๐ ๐,๐๐๐ฆ applies to the daylight period only.
2.5 Final comments
This Chapter starts with the basic physics of radiation and then deals with the theory of radiation as it is
required to model canopy photosynthesis and energy balance, including evaporation and transpiration,
which are considered in the following Chapters and are central to the growth of plants. The distinction
between shortwave and longwave radiation is crucial in the study of photosynthesis and energy dynamics.
The visible component of shortwave radiation is the source of energy for photosynthesis, and is known as
photosynthetic photon flux (PPF). The direct and diffuse components of PPF have been considered, and are
used in the description of canopy photosynthesis. The overall canopy energy balance includes both
shortwave and longwave radiation, and care must be taken to ensure that these components are described
appropriately. Direct measurements of longwave radiation are often not available and a simple approach
for estimating longwave radiation has been described, along with the overall canopy energy balance, in
relation to incoming shortwave radiation, air temperature and relative humidity.
2.6 References
Allen RG, Pereira LS, Raes D and Smith M (1998). FAO irrigation and drainage paper no. 56: crop
evapotranspiration. www.kimberly.uidaho.edu/ref-et/fao56.pdf.
Campbell GS (1977). An introduction to environmental biophysics. Springer-Verlag, New York, USA.
Campbell GS and Norman JM (1998). An Introduction to environmental biophysics, second edition.
Springer, New York, USA.
DairyMod and the SGS Pasture Model documentation 38
Jeffrey SG, Carter JO, Moodie KB and Beswick AR (2001) Using spatial interpolation to construct a
comprehensive archive of Australian climate data. Environmental Modelling & Software 16, 309โ330.
Johnson IR (2013). PlantMod: exploring the physiology of plant canopies. IMJ Software, Dorrigo, NSW
2453, Australia. www.imj.com.au/software/plantmod.
Johnson IR, Riha SG and Wilks DS (1995). Modelling daily net canopy photosynthesis and its adaptation to
irradiance and atmospheric CO2 concentration. Agricultural Systems 50, 1-35.
Jones HG (1992). Plants and microclimate. Cambridge University Press, Cambridge, UK.
Monteith JL & Unsworth MH (2008). Principles of environmental physics, third edition. Elsevier, Oxford,
UK.
Norman JM (1980). Interfacing leaf and canopy light interception models. In: Predicting photosynthesis for
ecosystem models, (Eds JD Hesketh and JW Jones), CRC Press, Boca Raton, Fl, USA.
Norman JM (1982). Simulation of microclimates. In Biometeorology in integrated pest management.
Academic Press, New York, USA.
Stockle CO & Campbell GS (1985). A simulation model for predicting effect of water stress on yield: an
example using corn. Advances in Irrigation, 3, 283-323.
Thornley JHM (2002). Instantaneous canopy photosynthesis: analytical expressions for sun and shade
leaves based on exponential light decay down the canopy and an acclimatized non-rectangular
hyperbola for leaf photosynthesis. Annals of Botany, 89, 451-458.
Thornley JHM and France J (2007). Mathematical models in agriculture. CAB International, Wallingford, UK.
Thornley JHM & Johnson IR (2000). Plant and crop modelling. Blackburn Press, Caldwell, New Jersey, USA.
Chapter 3: Pasture and crop growth 39
3 Pasture and Crop Growth
3.1 Introduction
Pasture growth and utilisation by grazing animals is central to the model and, in addition, forage crops are
also included which can be grazed as well as cut as conserved feed. The pasture module originates from
the general structure of the models described by Johnson and Thornley (1983, 1985); Johnson and Parsons
(1985); and Parsons, Johnson and Harvey (1988), although a number of modifications have been made.
More recent discussions of physiological pasture growth models can be found in Thornley and Johnson
(2000) and Thornley and France (2007). In addition, the approach has been developed in order to
incorporate water and nutrient effects. Cereal, brassica and bulb crops are implemented as developments
of the pasture model.
The following key points apply:
The model is constructed for generic species so that particular species are defined through the
basic model parameters.
The model includes carbon assimilation through photosynthesis and respiration followed by tissue
growth, turnover and senescence.
Plant growth and tissue dynamics are influenced by environmental conditions (light, temperature
and atmospheric CO2 concentration) as well as soil water and nutrient status.
Nitrogen dynamics and influence on growth are incorporated.
For annual pasture species, vegetative (emergence to anthesis) and reproductive (anthesis to
maturity) growth phases are included.
For cereal crops, phases characterised by vegetative growth, stem elongation, booting, anthesis,
soft-dough and maturity are included.
Brassicas are treated in a more simple manner than cereals, with vegetative and reproductive
phases included.
Multiple pasture species can be included, which may be perennial, annual, legume, C3 or C4.
Plant utilisation by grazing animals is considered in Chapter 6.
Plant digestibility and metabolic energy content is calculated in terms of the plant nutrient status,
although this is described in Chapter 6.
Throughout the discussion, the area of ground that is used is m2 although frequently in the
interface the hectare, ha, is used. The choice is generally to give the user access to familiar units
and should not cause any difficulty.
Photosynthesis calculations use moles CO2, which is converted to carbon units.
Carbon is the internal unit of plant mass used in the model and this can be converted to dry weight,
with conversion factor 0.45 kg C (kg d.wt)-1 as discussed in section 1.5 in Chapter 1.
Unlike the original models referenced above, the model described here does not include specific substrate
pools for labile carbon and nitrogen. Instead, in order to simplify the model, the daily carbon assimilation
and respiratory costs are calculated and the net carbon balance is then directly available for growth on that
day. In addition, the effect of available water and nutrients, as well as the influence of actual plant nutrient
status on growth are included.
Growth is calculated as follows:
The daily transpiration rate and the effect of water stress are calculated;
DairyMod and the SGS Pasture Model documentation 40
Daily photosynthesis is calculated in response to light, temperature, atmospheric CO2
concentration, canopy architecture, available water, and leaf nitrogen status;
Potential nutrient uptake is calculated in relation to root distribution and soil nutrient status;
Plant mass flux is calculated, incorporating tissue turnover, senescence, shoot and root growth;
Other processes, such as species interaction, nitrogen fixation (in legumes) are also included.
3.2 Transpiration and the influence of water stress
Potential transpiration, ๐ธ๐,๐๐๐ก, is the transpiration rate that occurs when there is no limitation due to
available soil water content, and is calculated according to the Penman-Monteith equation as discussed in
detail in Chapter 4. Potential transpiration is derived for full ground cover, so the actual transpiration
demand is given by
, ,T demand g T potE f E (3.1)
where ๐๐ is the live ground cover and is given by eqn (2.24) in Chapter 2.
Once transpiration demand is known it is then necessary to calculate the impact of soil water status and so
the actual transpiration. First, the growth limiting factor for water, ๐๐ค๐๐ก๐๐, is defined in relation to the
available soil water as a function of wilting point (๐๐ค), recharge point (๐๐), field capacity (๐๐๐), and
saturated water content (๐๐ ๐๐ก), as shown in Fig. 3.1. If ๐๐ค๐๐ก๐๐ is 1 then there is no limitation to growth; if it
is zero then there is total limitation. ๐๐ is the soil water content below which transpiration is reduced as a
result of limited available soil water. Field capacity and saturated water content are discussed in more
detail in Chapter 4.
Figure 3.1: Schematic representation of the influence of limiting soil water content
on transpiration.
For water contents below the wilting point, plants cannot extract water from the soil. Between the wilting
point and recharge point, ๐๐ค๐๐ก๐๐ increases from 0 to 1. Between recharge point and field capacity, ๐๐ค๐๐ก๐๐
is 1. Between field capacity and saturation, ๐๐ค๐๐ก๐๐ may decline, although this can be defined by the user.
The reason the ๐๐ค๐๐ก๐๐ can decline at soil water contents greater than field capacity is that plants may be
susceptible to water logging. Note that wilting point, field capacity and saturation are defined in the soil
water module of the interface, while the recharge point and any decline in the ๐๐ค๐๐ก๐๐ at saturation are
defined in the pasture or crop module. The term recharge point is used as this is the point at which
irrigation would have to be applied in order to prevent any water stress.
The strategy for calculating transpiration is to calculate ๐๐ค๐๐ก๐๐ for each soil layer, ๐๐ค๐๐ก๐๐,โ according to the
scheme illustrated in Fig. 3.1. The water uptake from each layer is then given by
Chapter 3: Pasture and crop growth 41
, , , ,T r water T demandE f g E (3.2)
where ๐๐,โ is the root fraction in each layer, so that the total transpiration is
1
,
totL
T TE E
(3.3)
where ๐ฟ๐๐๐ก is the total number of soil layers.
If there is no limitation to water uptake from any layer due to available soil water then water uptake
through the profile is taken out according to the relative root distribution. As water becomes unavailable
from layers, uptake from those layers is reduced according to ๐๐ค๐๐ก๐๐,โ.
According to eqn (3.3) there is no compensation for water limitation in dry layers by other layers that might
have abundant water. To allow this situation, the transpiration routines are run three times, or until
demand is satisfied. As long as the routines are run more than once, the results are relatively insensitive to
how many times they are repeated. Thus, eqn (3.3) with (3.2) becomes
3
1 1
, , , , ,
totL
T r water i T demand ii
E f g E
(3.4)
๐๐ค๐๐ก๐๐,โ,๐ is evaluated for each calculation loop, and ๐ธ๐,๐๐๐๐๐๐,๐ is reset for each loop to allow for
cumulative transpiration. The choice of three has been selected as giving appropriate responses for plant
growth for a wide range of locations.
Once the transpiration is known, the overall water growth limiting factor is defined as
,
Twater
T demand
E
E (3.5)
This is a useful indicator of water stress, and is also used in the calculations for partitioning growth between
shoots and roots.
3.3 Canopy photosynthesis
The calculations for daily canopy photosynthetic rate lie at the heart of this model as this is the source of
carbon for the whole system (apart from any imported supplementary feed). The canopy photosynthesis
component of the model is based directly on Johnson et al. (2010) and so the details are kept fairly brief
here. The source of energy for photosynthesis is the visible component of solar radiation, which was
discussed in Chapter 2 (section 2.4), and is referred to as photosynthetically active radiation (PAR) with
units J m-2 s-1, or photosynthetic photon flux (PPF) with units ยตmol photons: PPF is the standard
terminology in the plant physiology literature, and will be used here.
The strategy for calculating daily canopy photosynthesis is:
Define the instantaneous rate of leaf gross photosynthesis in response to PPF, temperature,
atmospheric CO2, leaf N;
Define light interception and attenuation through the canopy, which includes direct and diffuse PPF
components;
Integrate through the canopy to get canopy instantaneous gross photosynthesis;
Integrate through the day to get daily canopy gross photosynthesis;
Calculate the daily growth and maintenance
DairyMod and the SGS Pasture Model documentation 42
Combine gross photosynthesis and respiration to get daily net photosynthesis, which is the net
carbon assimilation by the canopy.
3.3.1 Units
The conventional units for leaf photosynthetic rate are ยตmol CO2 (m2 leaf)-1 s-1 whereas, for crop or pasture
growth rates, these are usually prescribed as kg d.wt. ha-1 d-1. Accordingly, in this model the leaf and
canopy photosynthetic rates use ยตmol CO2 for instantaneous or mol CO2 for per second or per day rates
respectively, and the daily canopy photosynthetic rate is then converted to carbon units, using eqn (1.69)
as discussed in Chapter 1, Section 1.5, which is then readily converted to d.wt units using eqn (1.73).
3.3.2 Leaf gross photosynthesis
The rate of single leaf photosynthesis, ๐โ ยตmol CO2 (m2 leaf)-1 s-1, in response to incident PPF, ยตmol photons
(m2 leaf)-1 s-1 is described by the non-rectangular hyperbola. This equation is discussed in detail in Chapter
1 (Section 1.3.2) and the equation for ๐โ can be written as
2 0m mP I P P I P (3.6)
where the parameters are:
๐๐ rate of single leaf gross
photosynthesis at saturating PPF
ยตmol CO2 (m2 leaf)-1 s-1
๐ผ leaf photosynthetic efficiency mol CO2 (mol photons)-1
๐ curvature parameter (dimensionless)
๐โ is given by the lower root of eqn (3.6), which is
1 2
214
2m m mP I P I P I P
(3.7)
Equation (3.7) is shown in Fig. 3.2
Figure 3.2: Leaf gross photosynthesis for ๐๐ =16 ยตmol CO2 (m2 leaf)-1 s-1,
๐ผ = 80 m mol CO2 (mol photons)-1, ๐= 0.8.
The influence of temperature, CO2 and nitrogen level on leaf gross photosynthesis is dominated by the
effect on the parameter ๐๐ in eqn (3.7). The quantum efficiency ๐ผ also depends on temperature and CO2,
although to a lesser extent than ๐๐. There is less evidence that the curvature parameter ๐ responds to
0 500 1000 1500 2000
PPF ( ยตmol / m2 ) / s )
0
5
10
15
20
( ยต
mo
l C
O2
/ m
2 )
/ s
Leaf gross photosynthesis
Chapter 3: Pasture and crop growth 43
these factors (Sands, 1995; Cannell and Thornley, 1998) and so this parameter is treated as constant. The
methods used here follow, or are adapted from, Cannell and Thornley (1998), Thornley (1998), and
Thornley and France (2007). The overall leaf photosynthetic response to the interaction between PPF,
temperature and CO2 is consistent with general observations in the literature: for more discussion, see
Johnson et al. (2010).
Light saturated photosynthesis, ๐ท๐
The general characteristics of the response of ๐๐ to temperature, CO2 concentration, and protein
concentration are:
๐๐ increases from zero as temperature increases from some low value;
There is an optimum temperature above which there is no further increase;
The temperature optimum increases in response to atmospheric CO2 concentration, ๐ถ, which is due
to the fall in photorespiration;
As temperature continues to rise there is a decline in ๐๐ for C3 species, also due to the increase in
photorespiration;
For C4 species, ๐๐ may remain stable or may decline slightly as temperature increases past the
optimum.
For C3 species, ๐๐ increases in response to increasing ๐ถ in an asymptotic manner, approaching a
maximum value at saturating ๐ถ.
C4 species show little photosynthetic response to increasing ๐ถ above ambient, ๐ถ๐๐๐,
๐๐ increases as the photosynthetic enzyme concentration increases, and it is assumed that this
enzyme concentration is proportional to the nitrogen content.
N content is expressed on a mass basis, kg N (kg C)-1
These factors are incorporated by defining
, , ,N,m m ref C Pm TC Pm NP P f C f T C f f (3.8)
where ๐๐ถ(๐ถ) is a CO2 response function, ๐๐๐,๐๐ถ (T,C) is a combined response to temperature and CO2,
๐๐๐,๐ is the response to protein concentration as related to N, ๐๐ kg N (kg C)-1, and ๐๐,๐๐๐ is a reference
value for ๐๐, and is the value of ๐๐ at a reference temperature, ๐๐๐๐, ambient CO2 concentration, ๐ถ๐๐๐,
and reference N concentration, as discussed below. The functions are constrained by
1, ,N ,,C amb Pm TC ref amb Pm N N reff C C f T T C C f f f (3.9)
Default values for ๐๐,๐๐๐ are
1
1
16
20
3 ,
4 ,
C : mol mol
C : mol mol
m ref
m ref
P
P (3.10)
which are taken to be representative of photosynthetic capacity within the canopy. However, it must be
noted that leaf photosynthetic potential is subject to considerable variation.
The CO2 response function, ๐๐ถ(๐ถ), is described in detail in Chapter 1 and is not discussed further here,
other than to note that
1C ambf C C (3.11)
and that the response is parameterised by defining the parameters ๐ and ๐๐ถ,๐ where
DairyMod and the SGS Pasture Model documentation 44
2
,
C amb
C C m
f C C
f C f
(3.12)
Default values are
1 2 1 5
1 05 1 1
3 ,
4 ,
C : . ; .
C : . ; .
C m
C m
f
f
(3.13)
According to these values, ๐๐ถ(๐ถ) increases by 20% and 50% when CO2 is double ambient and saturating
respectively, while for C4 plants the corresponding increases are 5% and 10%, which are lower than for C3
due to the lack of photorespiration and subsequent limited impact of increasing CO2 on the photosynthetic
capacity of C4 plants.
The N response function is the same for both C3 and C4 species and is defined as a simple ramp function, so
that
, ,
,N, , ,
,
,
N N ref N N refPm N
N mx N ref N N mx
f f f ff f
f f f f
(3.14)
According to this function, ๐๐๐,๐ increases linearly as the N concentration increases to the maximum value,
above which there is no further increase in the rate of photosynthesis.
Although N concentration is expressed in terms of plant carbon, The default parameter values
1
1
0 04
0 03
3 , ,
4 , ,
.C : kg N kg C
.C : kg N kg C
N ref N mxC
N ref N mxC
f fF
f fF
(3.15)
are used, where ๐น๐ถ is the carbon fraction of dry weight which is taken to be 0.45 kg C (kg d.wt)-1, as given
by eqn (1.69) in Chapter 1, Section 1.5. The values 0.04 and 0.03 correspond to 4% and 3% respectively in
the familiar units of N content as a percentage of dry weight. The lower
values for C4 plants reflects the
fact that these species generally have lower nitrogen concentration than C3.
The basic generic temperature response function that was described in Chapter 1, section 1.3.5 is used,
that is
0
1
1
0
,
,
mn
qopt mnmn
T mn mxr mn opt mn r
mx
T T
q T T qTT Tf T T T T
T T q T T qT
T T
(3.16)
where ๐๐ is a reference temperature, so that
1T rf T (3.17)
and ๐๐๐ฅ is given by
1 opt mn
mx
q T TT
q
(3.18)
The function takes its maximum value at ๐๐๐๐ก and is zero outside the range ๐๐๐ to ๐๐๐ฅ.
Chapter 3: Pasture and crop growth 45
The optimum temperature for photosynthesis is seen to increase in response to atmospheric CO2
concentration and so the combined ๐ and ๐ถ function, ๐๐๐,๐๐ถ(๐, ๐ถ), uses eqn (3.16), but with ๐๐๐๐ก defined
by
1, , ,opt Pm opt Pm amb Pm CT T f C (3.19)
where ๐๐ถ(๐ถ) is again given by the CO2 response function described in Chapter 1, and the parameter ๐พ๐๐
has default value
10 CPm (3.20)
C3 and C4 species are treated in the same way, with the exception that for C4 species the constraint
4 , , , ,C : , , , for Pm TC Pm TC opt Pm opt pmf T C f T C T T (3.21)
applies, so that the temperature response does not fall when temperatures exceed the optimum. The
decline in photosynthesis for C3 plants at high temperature is due to the shift from photosynthesis to
photorespiration, while, for C4 plants, photorespiration is generally negligible. In practice, there may be a
decline in photosynthesis at high temperatures due to water stress. Also, since, as discussed below,
respiration does increase with temperature, there will be a decline in net photosynthesis at high
temperatures for C4 plants.
Default values
20 3 23
25 12 35
3 , ,
4 , ,
C : C, C, C
C : C, C, C
ref mn opt Pm amb
ref mn opt Pm amb
T T T
T T T
(3.22)
are used, although these may vary for different species.
Photosynthetic efficiency, ๐ถ
Now consider the leaf photosynthetic efficiency, ๐ผ, which is defined by
3 15
4 15
, , , ,
, , ,
C : ,
C :
amb C TC N N
amb C N N
f C f T C f f
f C f f
(3.23)
where ๐ผ๐๐๐,15 mol CO2 (mol photons)-1 is the value of ๐ผ at ambient CO2 concentration, ๐ถ๐๐๐, and 15โฐC,
with default value
1
15 250, mmol CO mol photonsamb
(3.24)
The function ๐๐ผ,๐ถ(๐ถ) in eqn (3.23) captures the direct influence of ๐ถ on ๐ผ and is given by the same generic
response function that is used for ๐๐ in eqn (3.8).
The function ๐๐ผ,๐๐ถ(๐, ๐ถ) in eqn (3.23)defines the temperature response on ๐ผ and the influence of ๐ถ on this
response as given by
1
1
, ,,
,
,,
,
ambopt opt
TC
opt
CT T T T
f T C C
T T
(3.25)
where ๐ is a constant and
DairyMod and the SGS Pasture Model documentation 46
15 1,opt CT f C (3.26)
where, again, the generic CO2 response function is used. Note that eqn (3.25) will not be valid for very
small values ๐ถ as the term ๐ถ๐๐๐ ๐ถโ will become infinitely large. Rather than address this issue to deal with
unrealistic CO2 concentrations, the theory is restricted to CO2 concentrations greater than 100 ยตmol mol-1,
and subject to
0, , for all and TCf T C T C (3.27)
Default parameter values are
0 02. C and 6 C (3.28)
With these values, ๐๐๐๐ก,๐ผ increases from its ambient value of 15โฐC by 3โฐC for a doubling of CO2 from
ambient.
The function for ๐๐ผ,๐๐ defines the protein response for ๐ผ and is assumed to be a simple ramp function:
0 5 0 5
1
, ,,
,
. . ,
,
N N ref N N refN N
N N ref
f f f ff f
f f
(3.29)
where the values for ๐๐,๐๐๐ are given in eqn (3.15). This equation will not be valid for very low ๐๐ but, for
that situation, photosynthesis will be primarily restricted by the influence on ๐๐.
According to these equations, photosynthetic efficiency ๐ผ increases with increasing ๐ถ for both C3 and C4
species, but for C3 plants there is also a decline for temperatures above 15ยฐC. The increase in ๐ผ in response
to ๐ถ reflects the greater availability of CO2, while the decline in response to temperature for C3 species
indicates a shift towards photorespiration as temperature increases, while this shift is reduced at increasing
๐ถ. The lack of temperature response for C4 species is due to the lack of photorespiration in those plants.
The curves are not illustrated in detail here as they can be explored in DairyMod and the SGS Pasture
Model and, for more detail, in PlantMod (Johnson 2013). Figure 3.3 shows the leaf gross photosynthetic
response for C3 and C4 leaves at their reference N concentration and for ๐ = 20, 30โฐ and ๐ถ = ๐ถ๐๐๐ , 2๐ถ๐๐๐,
where ๐ถ๐๐๐ = 380 ppm. It can be seen that there is little impact of elevated CO2 on C4 photosynthesis,
whereas the response is quite noticeable for C3 plants. Also, while C4 photosynthesis is quite high for C4
plants at 30โฐC, it should be noted that high temperatures are often associated with significant water stress
and so these rates may not occur very often in practice. Indeed, possible direct benefits to elevated CO2
are likely to be offset by corresponding impacts due to temperature and water stress.
Chapter 3: Pasture and crop growth 47
Figure 3.3. Leaf gross photosynthetic response for C3 and C4 leaves at 20โฐC and 30โฐC as
indicated and at ambient CO2, left, and double ambient CO2, right.
3.3.3 Instantaneous canopy gross photosynthesis
The rate of instantaneous canopy gross photosynthesis, ๐๐ ๐mol CO2 (m-2 ground) s-1, is calculated by
summing the leaf photosynthetic rate over all leaves in the canopy, and is given by
0
L
gP P I d (3.30)
where ๐โ ๐mol CO2 (m-2 leaf) s-1, is the rate of leaf gross photosynthesis as discussed above, and
๐ผโ ๐mol photons (m2 leaf)-1 s-1, is the photosynthetic photon flux (PPF) incident on the leaf (Chapter 2), ๐ฟ
(m2 leaf) (m-2 ground) is the total canopy leaf area index, and โ is a dummy variable defining the cumulative
leaf area index through the depth of the canopy.
In Chapter 2 the direct and diffuse components of PPF were discussed. It is important to account for these
components since, as seen earlier, the leaf photosynthetic response to PPF is non-linear and so taking the
average PPF rather than separate direct and diffuse components can lead to an over-estimate of canopy
photosynthesis. For more discussion see Johnson et al. (2010) and Johnson (2013).
Separating the leaves into those in direct and diffuse PPF, eqn (3.30) can be written
0 0
, ,d ds dL L
g s s d dP P I P I (3.31)
which, using eqns (2.19) and (2.20) in Chapter 2 for โ๐ and โ๐ becomes
0 0
1, ,d d
L Lk k
g s dP P I e P I e (3.32)
Equation (3.32) is the key equation for calculating the rate of canopy gross photosynthesis which, combined
with the previous theory, incorporates the effects of PPF, temperature, leaf nitrogen, atmospheric CO2
concentration and total leaf area index.
The integrals in eqn (3.32) are difficult to solve analytically although quite straightforward to solve
numerically by summing through the canopy according to
0
5
10
15
20
25
30
0 500 1000 1500 2000
Pm
, ยตm
ol m
-2 s
-1
PPF, ยตmol photons m-2
0
5
10
15
20
25
30
0 1000 2000
Pm
, ยตm
ol m
-2 s
-1
PPF, ยตmol photons m-2
C3, T=20
C3, T=30
C4, T=20
C4, T=30
DairyMod and the SGS Pasture Model documentation 48
1
1, ,i i
i i
i nk k
g s di
P P I e P I e
(3.33)
where
1 2 1 12 2
, toi i i i n
(3.34)
and
L
n
(3.35)
According to this scheme, the canopy is divided into layers of depth ฮโ and ๐๐(๐ฟ) is evaluated at the mid-
point of each layer and the total enzyme content of the layer is this value multiplied by the layer depth.
This is a common scheme for numerical integration and, while more elaborate numerical techniques can be
applied, it works well for the present purposes. The value
0 1. (3.36)
is used throughout.
3.3.4 Daily canopy gross photosynthesis
The daily canopy gross photosynthesis, ๐๐,๐๐๐ฆ kg C (m-2 ground) d-1 is given by the integral of ๐๐ throughout
the day:
6
0
0 012 10, . dg day gP P t
(3.37)
where ๐ก is time (s), ๐ (s) is the daylight period in seconds, the factor 10โ6 converts from ๐mol CO2 to mol
CO2, and the factor 0.012 converts from mol CO2 to kg CO2. This equation can be applied with any daily
distribution of PPF and temperature. For constant PPF, ๐ผ0, and temperature, ๐, it is
600 012 10, . ,g day gP P I T (3.38)
where ฮ๐ก is a small time-step,
1 2 1 12 2
, toit t
t i t i i n
(3.39)
and
nt
(3.40)
Essentially, this scheme sums ๐๐ as evaluated at regular intervals throughout the day. The accuracy of the
numerical scheme will increase as the time step (ฮ๐ก) gets smaller, or the number of time steps (๐) gets
larger, although the computation will take longer. However, continuing to decrease ฮ๐ก to very small values
can cause numerical errors to increase and the scheme actually becomes less accurate. A general strategy
is to start with a relatively small value for ๐ and with ฮ๐ก calculated from eqn (3.40), gradually increase ๐
until the estimate of ๐๐,๐๐๐ฆ in eqn (3.38) starts to change. This sets a lower limit on ๐. In the present
model, the mean daytime PPF and temperature values are used and so eqn (3.38) applies.
Chapter 3: Pasture and crop growth 49
3.3.5 Daily canopy respiration rate
It is now necessary to calculate the daily respiration rate. Respiration, excluding photorespiration (which is
incorporated directly into the calculation of gross photosynthesis) is calculated using the McCree (1970)
approach, that has been further developed by Thornley (1970), Johnson (1990), and is widely used. This
identifies the growth and maintenance components of respiration. These components are helpful in
understanding the respiratory demand by the plants, although the actual underlying respiratory process
whereby ATP is produced from sugars with a respiratory efflux of CO2 is common to both growth and
maintenance respiration. Growth respiration is the respiration associated with the synthesis of new plant
material, while maintenance is the respiration required primarily to provide energy for the re-synthesis of
degraded proteins. Consequently, growth respiration is related to the growth rate of the plant, or daily
carbon assimilation, whereas maintenance respiration is proportional to the plant dry weight or, more
specifically, the actual protein content which may vary in response to plant nutrient status, particularly
nitrogen. For a background on this treatment of respiration, see Johnson (1990), Thornley and Johnson
(2000), Johnson (2013). The respiratory costs of nitrogen (N) uptake and N fixation are also incorporated.
Maintenance respiration
Maintenance respiration is generally regarded to be related to the plant live dry weight. However,
maintenance respiration is primarily related to the resynthesis of degraded proteins. There are other
maintenance costs, such as the energy required for phloem loading, but these are not considered explicitly,
so that it is assumed that enzyme concentration is an indicator of overall maintenance costs. In addition, as
a rate process, it is strongly temperature dependent. Incorporating these features, the maintenance
respiration is assumed to be given by
,,
Nm day ref m
N ref
fR m f T W
f (3.41)
where ๐๐(๐) is a maintenance temperature response function which takes the value unity at the reference
temperature ๐๐๐๐, ๐ (kg C m-2) is shoot mass, ๐๐ is the canopy N concentration kg N (kg C)-1 as used above
in the discussion of the influence of protein on the light saturated rate of leaf gross photosynthesis, ๐๐,
๐๐,๐๐๐ is the reference N concentration, and ๐๐๐๐ (d-1) is the maintenance coefficient at the reference
temperature and N content, with default value
10 025. drefm (3.42)
The maintenance temperature response function, ๐๐(๐), is defined to take the value unity at the reference
temperature ๐๐๐๐, so that
1m reff T T (3.43)
A simple linear response is used, as given by
,
,
m mnm
ref m mn
T Tf T
T T
(3.44)
with default values
3
12
3
4
C : C
C : C
mn
mn
T
T
(3.45)
and the same reference temperature as in eqn (3.22).
DairyMod and the SGS Pasture Model documentation 50
Growth respiration
According to the standard theory for the definition of growth respiration, one unit of substrate that is
utilised for growth results in ๐ units of plant structural material and (1 โ ๐) units of respiration, where ๐ is
the growth efficiency. Thus, for 1 unit of growth, this can be represented by the scheme in Fig. 3.4:
Figure 3.4: Schematic representation of growth respiration.
Thus, for growth ๐บ kg C m-2 d-1, the corresponding growth respiration is
1
gY
R GY
(3.46)
The respiratory costs for cell wall and protein synthesis are different, with the costs of the more complex
protein molecules being greater โ a detailed discussion can be found in Thornley and Johnson (2000). The
plant composition components discussed in Section 1.5 in Chapter 1, are used, so that the plant structure
comprises cell wall, protein and sugars, with molar concentrations ๐๐ค, ๐๐ and ๐๐ respectively, and where
1w p sf f f (3.47)
It is readily shown that, if the growth efficiencies for cell wall and protein are ๐๐ค, and ๐๐, then these are
related to the overall growth efficiency, ๐, by
111 pw
w pw p
YYYf f
Y Y Y
(3.48)
from which
1
111
pww p
w p
YYY
f fY Y
(3.49)
This allows for the direct influence of plant structure on the overall growth efficiency directly. The model
defaults are:
0.85; 0.55w pY Y (3.50)
so that, for example, with 20% sugars, 25% protein and 55% cell wall (on a mole basis), ๐ = 0.81, whereas,
if the protein content is reduced to 20% and the cell wall increased to 60%, this becomes ๐ = 0.83. Values
for ๐ that are observed experimentally are generally in the range 0.75 to 0.85. For more discussion, see
Johnson (1990), Thornley and Johnson (2000), Thornley and France (2007), Johnson (2013).
Chapter 3: Pasture and crop growth 51
Respiratory cost of nitrogen uptake and fixation
Nitrogen uptake involves energy costs (eg Johnson, 1990) and it is assumed that this is given by ๐ ๐,๐ข๐
where
, ,N up N up upR N (3.51)
where ๐๐ข๐ is the daily N uptake, kg N m-2 and ๐๐,๐ข๐, kg C (kg N)-1 is the N uptake respiration coefficient with
default value
1
0 6, . kg C kg NN up
(3.52)
The corresponding respiratory cost of N fixation in legumes is given by
, ,N fix N fix fixR N (3.53)
where ๐๐๐๐ฅ is the daily N fixation, kg N m-2 and ๐๐,๐๐๐ฅ, kg C (kg N)-1 is the N fixation respiration coefficient
with default value
1
6, kg C kg NN fix
(3.54)
The fact that ๐๐,๐๐๐ฅ is greater than ๐๐,๐ข๐ reflects the fact that the respiratory costs associated with N
fixation are significantly higher than for N uptake.
3.3.6 Daily carbon fixation
The previous section describes daily canopy gross photosynthesis and respiration. These are now
combined to give the daily carbon fixation, or net growth rate as given by
,g day g m NG P R R R (3.55)
where ๐ ๐ is the respiratory cost of N acquisition and, is
,
, ,
non-legumes:
legumes:
N N up
N N up N fix
R R
R R R
(3.56)
There is a circularity problem with the analysis here since growth depends on the respiratory cost of N
uptake, but this cost depends on growth. In order to avoid unnecessary complexity, and since this is a daily
time-step model, the value for ๐ ๐ from the previous day is used in the calculations.
Using eqn (3.46), eqn (3.55) becomes
,g day m NG Y P R R (3.57)
which defines the net plant growth rate, or carbon fixation, including roots.
3.3.7 Influence of temperature extremes on photosynthesis
The influence of temperature on leaf photosynthesis and respiration has been described above, but
temperature extremes may also affect photosynthetic capacity. For example, winter daytime temperatures
in southern Queensland may be suitable for a C4 species such as kikuyu, but low night temperatures may
prevent growth occurring. To accommodate this possibility, low and high cumulative temperature
functions can be defined and implemented in terms of the maximum and minimum daily temperatures.
DairyMod and the SGS Pasture Model documentation 52
First consider the low-temperature stress function. Two critical temperatures are defined, ๐๐๐,โ๐๐โ and
๐๐๐,๐๐๐ค, where ๐๐๐,โ๐๐โ is the critical temperature below which low-temperature stress will occur, and
๐๐๐,๐๐๐ค is the critical temperature at which the low-temperature stress is maximum. The low-temperature
stress function calculations are defined for situations where the minimum daily temperature, ๐๐๐, is either
greater than or less than ๐๐๐,โ๐๐โ
On day ๐, If ๐๐๐ < ๐๐๐,โ๐๐โ then the temperature stress coefficient is calculated as
0
,,
, ,, ,
,
,
,
mn mn lowmn mn low
mn high mn lowT low i
mn mn low
T TT T
T T
T T
(3.58)
which varies between 0 and 1 from ๐๐๐,๐๐๐ค to ๐๐๐,โ๐๐โ.
Conversely, if ๐๐๐ โฅ ๐๐๐,โ๐๐โ the recovery coefficient is calculated according to
, ,,
meanT low i
sum low
T
T (3.59)
where ๐๐ ๐ข๐,๐๐๐ค is a critical temperature sum for recovery from low temperature stress.
Starting with ๐๐,๐๐๐ค,0 = 1, either eqn (3.58) or (3.59) is calculated depending on the minimum daily
temperature. If ๐๐๐ < ๐๐๐,โ๐๐โ then the cumulative temperature stress function is calculated as
, , ,T low T low i (3.60)
whereas, if ๐๐๐ โฅ ๐๐๐,โ๐๐โ then the cumulative stress function is now calculated as
1, , , ,min ,T low T low T low i (3.61)
The daily gross photosynthesis, ๐๐,๐๐๐ฆ is then multiplied by ๐๐,๐๐๐ค to incorporate the influence of
cumulative low temperatures or recovery from low temperatures.
In practice, low temperature stress increases in response to a sequence of low temperature and then the
plant can recover as temperature increases. Full recovery from full stress, that is when ๐๐,๐๐๐ค is zero, will
occur when there are no further days with ๐๐๐ < ๐๐๐,โ๐๐โ and when the temperature sum of the mean
daily temperature exceeds ๐๐ ๐ข๐,๐๐๐ค.
The default parameter values are
0 5 100
3 7 100
3 , , ,
4 , , ,
C : C, C, C
C : C, C, C
mn low mn high sum low
mn low mn high sum low
T T T
T T T
(3.62)
although it should be noted that, by default, low temperature stress is not implemented for C3 plants.
The effect of high temperature stress is defined in an analogous way to low temperatures, but now the
critical temperature sum for recovery is defined by
0 25
, ,,
max , meanT high i
sum high
T
T
(3.63)
Default parameter values are
Chapter 3: Pasture and crop growth 53
35 30 100
38 35 100
3 , , ,
4 , , ,
C : C, C, C
C : C, C, C
mx high mx low sum high
mx high mx low sum high
T T T
T T T
(3.64)
although it should be noted that, by default, high temperature stress effects are not implemented.
Although this treatment of low and high temperature stresses on photosynthesis is completely empirical, it
captures the influence of temperature extremes and subsequent recovery.
3.4 Root distribution
The distribution of roots is important because of its influence on factors such as water and nutrient uptake
as well as the input to the soil organic matter. For vegetative species, root depth is taken to be constant,
whereas for annual species it increases to its maximum value at anthesis (flowering), as described later.
The relative root distribution by weight is shown in Fig 3.5, and is defined by:
,
1
1
rr q
r h
f z
z
d
(3.65)
where ๐๐,โ is the depth for 50% relative root mass and ๐๐ is a scaling parameter. This is a convenient
empirical approach, whereby ๐๐ = 0.5 when ๐ง = ๐๐,โ.
Root distribution has also been described according to an exponential equation, as used by Gerwitz and
Page (1974) when they analysed a large range of root distribution data. However, the data are very
variable and the sigmoidal pattern is probably preferable as it allows for a concentration of roots near the
surface, sometimes referred to as the plough layer.
Figure 3.5: Relative root distribution, and corresponding cumulative root distribution as a
function of depth using eqn (4.17), with ๐๐,โ = 25 cm, ๐๐=3,
and a total root depth of 100 cm.
3.5 Nitrogen remobilisation, uptake, and fixation
Nitrogen is available for growth from soil inorganic NO3 and NH4, ๐๐ข๐; N fixation in the case of legumes,
๐๐๐๐ฅ; and any remobilised N from senescent material, ๐๐๐๐๐๐, and can be written
avail up fix remobN N N N (3.66)
where all terms have units kg N m-2. For non-legumes, ๐๐๐๐ฅ is obviously zero.
0.00 0.20 0.40 0.60 0.80 1.00
Relative root distribution
100
80
60
40
20
0
De
pth
(cm
)
0.00 0.20 0.40 0.60 0.80 1.00
Cumulative root distribution
100
80
60
40
20
0
De
pth
(cm
)
DairyMod and the SGS Pasture Model documentation 54
Remobilisation of N is accounted for in the model by recycling N from senescent tissue, and is taken to be
proportional to senescence, and so is given by
,d
ddead
remob N remobgross
WN f
t (3.67)
where the derivative term is the gross production of dead material prior to, for example, losses from
standing dead to litter.
Nitrogen uptake from soil inorganic NO3 or NH4 are treated on a pro-rata basis. Thus, if the concentration
of inorganic nitrogen in any layer is [๐]โ, kg N (kg soil)-1, and the root fraction is ๐๐,โ, kg root C m-2, the
potential N uptake is
,up N rN N W (3.68)
where ๐๐, kg soil (kg root C)-1 d-1 is a nitrogen uptake coefficient. This parameter, while quite simple, is not
particularly intuitive to work with. In the model, it has been calculated in relation to reference available N
and root dry weight, with default value
1 1 1200 gN t root d.wtN ppmN d (3.69)
where ๐๐๐๐ is the soil N concentration expressed as ppm, or mg N (kg soil)-1. Thus, in these units, if the
soil N concentration is 10 ppm, 1 t roots ha-1 will take up 2 kg N.
N fixation in legumes is an important source of nitrogen in many crop and pasture systems. The treatment
of N fixation in the model is quite simple and is structured to ensure that legumes can obtain the required
N for the optimum N concentration in new growth. Thus,
0 ,max ,fix req opt remob upN N N N
(3.70)
So that N acquisition in legumes meets optimum requirement. It is apparent from this equation that if the
available N from remobilisation and uptake is sufficient for optimum demand then there will be no fixation.
Once the total N available for growth is known, it is possible to define a nitrogen limiting factor, analogous
to that for water (eqn (3.5)), as
1,
min , availN
req opt
N
N
(3.71)
which is used later when considering the partitioning of growth between shoots and roots.
3.6 Pasture growth, senescence and development
The analysis so far defines the carbon inputs through photosynthesis and respiration as well as nitrogen
remobilisation, uptake and, in the case of legumes, fixation. It now remains to describe overall canopy
growth in terms of shoot and root growth and senescence, as well as the leaf and sheath components of
shoot growth. The pasture growth model has been compared extensively with observed data for a range of
locations (for example, Cullen et al., 2008) and so discussions of actual model behaviour in different
locations is not discussed here.
A key feature of the treatment of pasture growth here is the turnover of plant tissue, which has been
shown to have a major impact on pasture growth and utilisation (eg Parsons, 1988), and is widely used.
The flow of tissue through the system is shown schematically in Fig. 3.6:
Chapter 3: Pasture and crop growth 55
Figure 3.6: Schematic representation of growth, tissue turnover and senescence.
According to this scheme there are leaf categories, including the associated sheath and stem,
corresponding to growing leaves, two categories of live leaves, and standing dead. New growth goes to the
growing leaf box. There is a flow of tissue through the categories until it is transferred to the litter. Carbon
is also partitioned to the roots, although separate root categories are not included. Root senescence
passes straight to the soil organic matter pool.
The net carbon assimilation was described in detail in the previous section and the dynamics of growth,
partitioning, tissue turnover and senescence are now considered.
First, the basic state variables in the model are define:
Live leaf: 1 3, , , tolive iW i (3.72)
Total live leaf: 3
1
, , ,live live ii
W W
(3.73)
Live sheath + stem 1, , , to3live s iW i (3.74)
Total live sheath+stem: 3
1
, , ,live s live s ii
W W
(3.75)
Total live shoot: , , ,live shoot live live sW W W (3.76)
Dead leaf ,deadW (3.77)
Dead sheath + stem ,dead sW (3.78)
Root live rW (3.79)
3.6.1 Shoot:root partitioning
Shoot:root partitioning is important in terms of the plantโs ability to access water and nutrients, its impact
on shoot, and therefore harvestable, growth, and also on the supply of organic matter to the soil for soil
organic matter and inorganic nutrient dynamics. Various models of shoot:root partitioning are discussed in
Thornley and Johnson (2000). The models partition growth in a way that attempts to balance the
requirements between resources acquired by the shoot and root respectively. For example, if water is
limiting then plants will partition a greater proportion of growth to the roots to attempt to increase water
uptake.
The partitioning of new growth to the shoot, ๐บ๐ โ๐๐๐ก kg C m-2 d-1, is defined as
Net carbonassimilation
Growingleaves,
sheath andstem
Live leaves,sheath and
stem
Live leaves,sheath and
stem
Standingdead
Roots
litter
Soil organic matter
DairyMod and the SGS Pasture Model documentation 56
1 2
,shoot shoot ref water NG G (3.80)
where the growth limiting factors, ฮฉ๐ค๐๐ก๐๐ and ฮฉ๐ are given by eqns (3.5) and (3.71), and ๐บ is the total
plant growth rate, eqn (3.57). According to this equation, the proportion of new growth partitioned to the
shoot declines as water or nitrogen stress increases. The square root term moderates the response and
has been seen to give realistic partitioning patterns. Note that for legumes, ฮฉ๐ always takes the value
unity.
Root growth, ๐บ๐๐๐๐ก kg C m-2 d-1, is then simply
root shootG G G (3.81)
3.6.2 Leaf:sheath partitioning and leaf growth
Once the proportion of shoot growth is defined by eqn (3.80) it is partitioned linearly between the leaf and
sheath, so that
shoot
s s shoot
G G
G G
(3.82)
where ๐บโ and ๐บ๐ , kg C m-2 d-1 are the daily rates of leaf and sheath growth.
Equation (3.82) defines the leaf mass growth, and it now remains to define the associated leaf area index
production. The specific leaf area, ๐ m2 leaf (kg C)-1, is defined as
L
W (3.83)
where ๐ฟ, m2 leaf (m2 ground)-1 is the leaf area index and ๐โ, kg C (m2 ground)-1 is the leaf mass. ๐ is seen to
depend on atmospheric CO2 concentration and, following Johnson et al. (2010), is taken to be
amb
Cf C
(3.84)
where ๐๐๐๐ is the value of ๐ at ambient CO2 and ๐๐ถ(๐ถ) is the generic CO2 response function used earlier in
the treatment of leaf photosynthesis, and discussed in Chapter 1, Section 1.3.4. ๐ is quite variable for
different species, but as an example and on a dry weight basis the default value for perennial ryegrass is
equivalent to 20 m2 leaf (kg d.wt)-1.
3.6.3 Growth dynamics
Figure 3.6 shows the scheme for tissue turnover and senescence. Define the flux parameter as ๐พ, d-1, (with
subscript dead for the flux from standing dead to litter), the flux of material between each category can be
defined as:
11 2 2 ,liveflux W (3.85)
22 3 ,liveflux W (3.86)
33 ,liveflux dead W (3.87)
dead deadflux dead litter W (3.88)
Chapter 3: Pasture and crop growth 57
The factor 2 in eqn (3.85) allows for the fact that the mean d.wt of leaves in the growing leaf category will
be about half of the fully grown leaves in that category. It is assumed that the flux between the live leaf
categories are lower for the sheath than the leaf.
The growth dynamics equations can now be written
Leaf
112
, ,, ,
d
d
liveshoot live
WG W
t (3.89)
21 22
, ,, , , ,
d
d
livelive live
WW W
t (3.90)
32 3
, ,, , , ,
d
d
livelive live
WW W
t (3.91)
3,
, , ,
d
d
deadlive dead dead
WW W
t (3.92)
Sheath + stem
111 2
, ,, ,
d
d
live sshoot s live s
WG W
t (3.93)
, ,2, ,1 , ,2
d2
d
live ss live s s live s
WW W
t (3.94)
32 3
, ,, , , ,
d
d
live ss live s s live s
WW W
t (3.95)
,, ,3 ,
d
d
dead ss live s dead dead s
WW W
t (3.96)
Root
d
dr
root r rW
G Wt
(3.97)
In the model, a scale factor is defined to relate ๐พ๐ and ๐พ๐ to ๐พโ although individual values could be
prescribed if necessary.
The rate constant, ๐พโ, can be related to the number of live leaves per tiller, number of boxes in Fig. 3.6, and
the leaf appearance rate, ๐โ d-1, and is
number of boxes
live leaves per tiller (3.98)
which can be helpful in defining the flux parameters. In the model, which has 3 leaf age category boxes,
the leaf appearance interval and live leaves per tiller are defined and, since the leaf appearance interval is
the reciprocal of leaf appearance rate, this defines ๐พโ.
DairyMod and the SGS Pasture Model documentation 58
Nitrogen
The equations for plant dynamics above apply to nitrogen as well as carbon, but now scale factors for N
concentration in non-leaf shoot material and root material are defined, which are both 0.5 by default.
Furthermore, remobilisation of N from senescent tissue is an additional flux into the growing leaf.
3.6.4 Influence of temperature on growth dynamics
The rate of leaf appearance is strongly temperature dependent and so the ๐พ rate constants are related to
mean daily temperature by
reff T (3.99)
where ๐๐พ(๐) again uses the temperature response function with an asymptote, eqn (1.41) discussed in
Chapter 1, Section 1.3.6. Default parameters for perennial ryegrass are
1 8 3 20 2; ; ;mn optT C T C q (3.100)
with this value of ๐โ corresponding to a leaf appearance interval of 8 d at 20โฐC. Both the leaf appearance
rate and interval are illustrated in Fig.3.7 for these parameter values.
Figure 3.7: Leaf appearance rate and interval for the default parameter values for
perennial ryegrass as given in eqn (3.100)
3.7 Mixed swards
The analysis so far has considered a single species and we now consider mixed swards, which involves
defining light interception and photosynthesis in mixtures, as well as allowing for root distribution of
multiple species.
3.7.1 Light interception and attenuation
The theory is based on Johnson et al. (1989) and Thornley and Johnson (2000) but is developed to allow for
canopies of different heights.
First, the relationship between canopy leaf area index, ๐ฟ, and height, โ, is defined using the non-
rectangular hyperbola discussed in Chapter 1, Section 1.3.2, which in terms of ๐ฟ and โ variables, can be
written
0 5 10 15 20 25
Mean daily temperature, ยฐC
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Da
ys / le
af
Leaf appearance rate
0 5 10 15 20 25
Mean daily temperature, ยฐC
0
10
20
30
40
Da
ys / le
af
Leaf appearance interval
Chapter 3: Pasture and crop growth 59
1 2
214
2m m mh L h L h h L
(3.101)
where, as discussed in Section 1.3.2, ๐ผ is the initial slope of the response, โ๐ the asymptote, which is the
maximum canopy height, and ๐ a curvature parameter between 0 and 1. This equation was used above for
the leaf photosynthetic light response curve. A characteristic of the present model is to try to express the
underlying equations in terms of parameters that have some underlying physiological interpretation. For
the relationship between โ(๐ฟ), the initial slope of the curve is not easy to estimate. However, by defining
the value of ๐ฟ for half-maximum height, ๐ฟโ๐๐๐, it is readily shown after some algebra that
2
2m
half
hL
(3.102)
so that, instead of prescribing ๐ผ, ๐ฟโ๐๐๐ can be defined instead. In the model, a fixed value for the curvature
parameter is used, which is
0 9. (3.103)
The maximum height, and consequently ๐ฟโ๐๐๐ are likely to vary significantly between species, and the
default values for perennial ryegrass are
1
2 20 5 2. m, m leaf m groundm halfh L
(3.104)
The response is shown in Fig. 3.8 for these parameter values.
Figure 3.8. Relationship between canopy height and LAI as given by eqns (3.101) to (3.104).
The procedure now is to adapt the light attenuation and interception theory that was described in Chapter
2, Section 2.4.5, and that was applied above for a single species. The procedure is numerical and uses the
LAI increment ๐ฟ๐ฟ where, for the calculations,
0 1.L (3.105)
and then sums through the canopy at these LAI increments to the total canopy LAI. Thus, the total number
of layers is
totL
Ln
L (3.106)
0.0 2.0 4.0 6.0 8.0 10.0
LAI
0
10
20
30
40
50
He
igh
t, c
m
Height vs LAI
DairyMod and the SGS Pasture Model documentation 60
where ๐ฟ๐ก๐๐ก is the sum of the LAI for each species. Each species is then assumed to be distributed through
the canopy on a pro-rata basis of their height. Thus, if the canopy height is โ๐๐ฅ, which is the height of the
species with maximum height, then the starting layer for each species is
1, roundmx p
start pmx
h hn
h
(3.107)
where ๐ denotes the species and โroundโ is a programming technique to round floating point values to
integer, since ๐๐ ๐ก๐๐๐ก,๐ must be integer. The LAI increment for each species now becomes
,
pp
L start p
LL
n n
(3.108)
Thus, we now know the LAI increments for each species and which layer, from the top, where they first
occur in the canopy.
The procedure is now to integrate through the canopy to calculate the light interception on each species in
each canopy layer. Following Johnson et al (1989), an โeffectiveโ light extinction coefficient for each layer is
defined as
,p p
e ip
k Lk
L
(3.109)
where the summations are over all species and a species is only included in the summation if
,start pn i (3.110)
Extending the theory in Chapter 2, Section 2.4.5, the direct and diffuse components of PPF in any layer,
denoted by ๐ are given by
1 1, ,s i s i e iJ J k L (3.111)
and
1 1, ,d i d i e iJ J k L (3.112)
respectively, where
i pL L (3.113)
is the total LAI in the layer and, again, a species is only included in the summation if eqn (3.110) applies.
Equations (3.111) and (3.112) are then used in the canopy photosynthetic integrals given by eqn (3.32) or,
more precisely the numerical procedure defined immediately after that equation.
3.7.2 Root distribution
Root distribution for each species was discussed above in Section 3.4. The uptake of both water and
nutrients for multiple species on any particular day is calculated for each species as if the other species
were not present. The rationale for this is that the relative amount of water or nutrients that is removed
from the soil on any one day is generally a small fraction of the total available. However, on the next day,
the pool of available soil resources will have been depleted by the amount used by all species. A minor
problem may be that if a large proportion of resources is removed on any one day then the order in which
species are treated may have a small effect on the simulation. In order to avoid any problems in the
Chapter 3: Pasture and crop growth 61
unlikely event that this occurs, each species is allocated a maximum possible amount of available water or
nitrogen in each soil layer which is based on their fractional root mass in that layer.
3.8 Crop growth
Crop growth is described by developing the theory for pasture species to include different growth phases.
Cereals and brassicas are treated separately, while forbes can be implemented directly from the pasture
component. As mentioned earlier, the pasture module has been compared extensively with experimental
data for a range of locations, whereas this is not the case for the crop module which is a relatively recent
addition to the model at the time of writing. Consequently, some actual model behaviour is presented in
this section of the documentation. There are parameter sets for a few crops in the model interface, and
these can be adapted for other crops. These include maize, which may be used as a generic crop to
parameterise other C4 crops. Maize is a warm-temperature crop and for some locations temperatures may
not be sufficient for the full growth cycle to be completed within a year of sowing.
3.1.1 Cereals
It is common in crop models to include a detailed description of the developmental phases โ for example,
the Zadocks scale is a 0-99 scale of development characterising different stages of growth. This degree of
complexity need only be used if necessary, for example, it may be adequate to define crop growth simply
as the germination, vegetative, reproductive and maturation phases.
For the present model, the key development stages relevant to grazed cereals are implemented, and so the
following phases are included:
Germination and vegetative growth โ leaf, sheath and root growth;
Stem elongation โ this is the point where stem elongation occurs and the growing point (meristem)
rises through the plant stem;
Booting โ the growing point emerges;
Anthesis โ flowering commences, new leaf growth ceases, root growth ceases;
Soft dough โ grain ripening starts
Maturity โ the crop can be harvested for grain.
The crop can be grazed prior to stem elongation and vegetative growth will continue. However, if it is
grazed after that stage, the growing points will be eaten and growth will therefore cease. A management
decision can be made so that the developmental cycle for that crop ceases when it is grazed at stem
elongation. Alternatively, it can be allowed to grow beyond that phase and then harvested for silage
between booting and soft dough. If it is grown to maturity then it is harvested as a grain crop.
Phase duration and vernalization
Phase duration is defined using the Tsum (temperature sum) hypothesis, which is accumulated day-degrees
above a specified temperature threshold (see, for example, Thornley and Johnson, Plant and Crop
Modelling, 2000). So, for example, if the mean daily temperature is 18 ยฐC and the base temperature 5 ยฐC,
then the contribution to the Tsum on that day is 13 ยฐC. In addition, vernalization is defined using an
accumulated temperature sum below an upper temperature. So, for example, if the upper temperature is
10 ยฐC and the minimum temperature on a particular day is 4 ยฐC, then the contribution to the vernalization
Tsum is 6 ยฐC.
It is challenging for users to specify Tsum parameters because of the limited availability of data. However,
using their experience with particular locations, they are often familiar with the approximate dates of
phase changes. In order to assist with defining these Tsum parameters, there is a feature in the model
DairyMod and the SGS Pasture Model documentation 62
interface that calculates average dates of phase changes for any climate file that has been loaded. These
dates are shown for winter wheat growing at Elliott and Terang, which have contrasting climate
characteristics. It should be emphasised that these dates are long-term averages and will vary between
years depending on local temperatures. Table 1 shows the average dates of phase change for the default
winter wheat parameter set for both Elliott and Terang.
Table 1. Average dates for phase change with the default winter wheat parameter set for Elliott (Tas) and Terang (Vic), sown on 15 March and emerging on 25 March.
Location Vernalize Stem elongation
Booting Anthesis Soft dough Maturity
Elliott 23 June 8 Sept 4 Oct 17 Nov 14 Dec 16 Jan
Terang 8 July 26 Aug 14 Sept 22 Oct 15 Nov 15 Dec
Growth characteristics during phases
Growth from emergence to booting and booting to anthesis is described identically to that for perennial
plants. However, if the plant is grazed at booting this will effectively halt growth since the growing point
will be grazed. During each phase, the cumulative Tsum is evaluated, and the phase-fraction is calculated
as
,
sumphase
sum crit
Tf
T (3.114)
where ๐๐ ๐ข๐ is the Tsum since the beginning of the phase and ๐๐ ๐ข๐,๐๐๐๐ก is the critical Tsum at which the
phase change occurs. So, for example, when ๐๐โ๐๐ ๐ reaches 1 during the emergence to stem elongation
phase, this triggers the switch to stem elongation and commencement of the stem elongation to booting
phase.
For physiological processes, the phases are combined to define vegetative and reproductive growth.
Vegetative growth includes the phases to anthesis, anthesis to soft dough and soft dough to maturity
defining reproductive growth.
During vegetative growth, the only effect of ๐๐โ๐๐ ๐ is to determine the root depth, which is given by
0 1 0 1 , , ,. .r depth r depth mx vegW W f (3.115)
where ๐๐ฃ๐๐ is ๐๐โ๐๐ ๐ for the vegetative phase. According to this simple equation, root depth starts at 0.1m,
or 10cm, and increases linearly to its maximum value.
During reproductive growth, there is a shift from leaf, sheath and stem growth to grain production. First, a
generic scale function is defined using the mathematical form of the temperature response function that
has an optimum, as defined in eqn (1.41), with the function increasing from zero to 1 as the ๐๐๐๐ (the ๐๐โ๐๐ ๐
value for reproductive growth) value increases from zero to 0.5, and taking the value 1 above ๐๐๐๐ = 0.5.
Denoting this scale factor by ๐, the partitioning of new growth between root, shoot, shoot components,
and grain is then evaluated.
First, the shoot fraction of new growth is calculated according to section 3.6.1 for vegetative growth.
Denoting this by ๐๐ โ๐๐๐ก,๐ฃ, the actual shoot fraction of new growth is
1 , ,shoot shoot v shoot v (3.116)
Chapter 3: Pasture and crop growth 63
so that is unity when ๐ = 1, at which time all root growth ceases.
Once the fraction of new growth to the shoot is defined, the associated fraction to grain is defined by by
grain shoot (3.117)
so that, as the reproductive approaches maturity, new growth is increasingly partitioned to grain.
As well as grain growth, there is a shift from leaf to stem growth. In this case, denoting the leaf fraction of
new shoot growth during vegetative growth as ๐โ,๐ฃ, as calculated in section 3.6.2, the actual leaf and stem
fractions of growth are
1 1 ,grain v (3.118)
1 s grain (3.119)
As well as calculating the growth fractions during reproductive growth, the flux of material through the live
categories, as illustrated in Fig. 3.6 and discussed in section 3.6, is also affected. In particular the flux
coefficients for leaf and stem, where stem is included in the sheath component, are modified according to
1 , ,v v (3.120)
1 , ,s s v s v (3.121)
1 , ,r r v r v (3.122)
Thus, as the reproductive phase approaches maturity, all live shoot material is transferred to the dead
components.
Simulations
Although cereal crops are often grazed or harvested for silage prior to maturity, it is instructive to see the
simulated crop yields, as shown in Fig. 3.9 again for Elliott and Terang. Simulations are run from 1971 to
2011 with non-limiting nitrogen; the crops are sown each year on 15 March and germination occurs on the
first day where the soil water content is at least 85% of field capacity, 10 days after sowing (these
parameters can, of course, be adjusted by the user). One immediate characteristic to observe is that, for
Elliot, the yields are very high in some years and for others they are zero. This is because the model is
reporting yields for calendar years where the crop may be mature in January and the next crop in
December of the same year. The average yields are 7.8 and 6.4 t ha-1 for Elliott and Terang respectively.
1970 1980 1990 2000 2010
0
5
10
15
20
( t / h
a )
/ y
r
Annual grain yield
1970 1980 1990 2000 2010
0.0
2.0
4.0
6.0
8.0
10.0
( t / h
a )
/ y
r
Annual grain yield
DairyMod and the SGS Pasture Model documentation 64
Figure 3.9. Annual grain yields (1971-2011) for winter wheat at Elliott (left) and Terang (right)
under non-limiting nitrogen.
The results in Fig. 3.9 are consistent with crop yields in these regions.
3.1.2 Brassicas
Brassicas in the model are either annual leafy crops or bulb crops, with the leafy crops being represented
by kale and chicory, and bulb crops by turnip. Again, parameter sets for these crops can be adapted to
implement other crops as required. As discussed in the Chapter 7 (Management), brassicas are only grazed
once in the model and so it is not necessary to include as many development phases as for cereals. The
phases are:
Germination
Anthesis
Maturity
However, in practice these crops will not be grown to maturity, being grazed much earlier during the
vegetative phase. Consequently, it is not particularly important in the model to parameterise the Tsum
requirements for anthesis and maturity. Nevertheless, these data have been included for the sake of
completeness: they are not presented here but are readily explored in the model.
Leafy brassicas
Using these phases, growth is then defined in an analogous manner to cereal crops, but without a grain
component, and so is not discussed further here.
Bulb crops
Bulb crops are treated in a similar manner to cereals, but with bulb growth occurring during vegetative
growth. This is treated analogous to grain growth described above, but with the ๐ scaling coefficient
calculated during the vegetative phase. The bulb fraction of new shoot growth is given by
0 5 .bulb (3.123)
so that the maximum proportion of new shoot growth partitioned to the bulb is 0.5 and this occurs as the
crop approaches anthesis. Following anthesis, there is no further bulb growth.
3.9 Concluding remarks
The model for pasture and crop growth described here is a physiologically based carbon assimilation model
in response to environmental conditions. Daily growth rate is estimated by starting with leaf
photosynthesis, summing this over the canopy and accumulating over the day. Dark respiration
components for maintenance, nitrogen uptake and, for legumes, nitrogen fixation are incorporated. In
addition, generic parameters for C3 and C4 pastures are included. Multiple pasture species interactions are
also incorporated. The model includes the effects of tissue turnover, phase development, annual and
perennial species as well as legumes. In addition, the nutrient composition of live and dead tissue is
calculated which is used both in the treatment of nutrient cycling and also animal nutrition. Water and
nutrient dynamics interact with the soil water and nutrient balances in a consistent manner. The model
was originally developed for pasture species but has been adapted to include cereal and brassica crops.
Chapter 3: Pasture and crop growth 65
3.10 References
Cannell MGR, Thornley JHM. (1998). Temperature and CO2 responses of leaf and canopy photosynthesis:
a clarification using the non-rectangular hyperbola model of photosynthesis. Annals of Botany, 82, 883-
892.
Cullen BR, Eckard RJ, Callow MN, Johnson IR, Chapman DF, Rawnsley RP, Garcia SC, White T, Snow VO
(2008) Simulating pasture growth rates in Australian and New Zealand grazing systems. Australian
Journal of Agricultural Research, 59, 761-768.
Gerwitz A and Page ER (1974). An empirical mathematical model to describe plant root systems. Journal of
Applied Ecology, 11, 773 โ 781.
Johnson IR (2013). PlantMod: exploring the physiology of plant canopies. IMJ Software, Dorrigo, NSW,
Australia. www.imj.com.au/software/plantmod.
Johnson IR, Thornley JHM, Frantz JM, Bugbee B (2010). A model of canopy photosynthesis incorporating
protein distribution through the canopy and its acclimation to light, temperature and CO2. Annals of
Botany, 106, 735-749.
Johnson IR (1990). Plant respiration in relation to growth, maintenance, ion uptake and nitrogen
assimilation. Plant, Cell and Environment, 13, 319-328.
Johnson IR, Parsons AJ and Ludlow MM (1989). Modelling photosynthesis in monocultures and mixtures.
Australian Journal of Plant Physiology, 16, 501-516.
Johnson IR and Thornley JHM (1983). Vegetative crop growth model incorporating leaf area expansion and
senescence, and applied to grass. Plant, Cell and Environment, 6, 721-729.
Johnson IR and Thornley JHM (1985). Dynamic model of the response of a vegetative grass crop to light,
temperature and nitrogen. Plant, Cell & Environment 8, 485โ499.
Johnson IR and Parsons AJ (1985). A theoretical analysis of grass growth under grazing. Journal of
Theoretical Biology, 112, 345-367.
McCree KJ (1970). An equation for the respiration of white clover plants grown under controlled
conditions. In Prediction and Measurement of Photosynthetic Productivity (ed. I. Setlik), pp 221 โ
229. Pudoc, Wageningen.
Parsons AJ (1988). The effects of season and management on the growth of grass swards. In: The Grass
Crop - the Physiological Basis of Production (eds MB Jones and A Lazenby). Chapman Hall, London, 243-
275.
Parsons AJ, Johnson IR and Harvey A (1988). The use of a model to optimise the interaction between the
frequency and severity of intermittent defoliation and to provide a fundamental comparison of the
continuous and intermittent defoliation of grass. Grass and Forage Science, 43, 49-59
Parsons AJ, Carrere P and Schwinning S (2000). Dynamics of heterogeneity in a grazed sward. In: Grassland
Ecophysiology and Grazing Ecology (eds G Lemaire, et al ). CAB International, Wallingford (UK).
Passioura JB and Stirzaker RJ (1993). Feedforward responses of plants to physically inhospitable soil.
International Crop Science I, pp 715 โ 719. Crop Science Society of America, Madison, WI.
Robson MJ and Sheehy JE (1981). Leaf area and Light Interception. In Sward Measurement Handbook (eds J
Hodgson, R D Baker, A Davies, A S Laidlaw and J D Leaver), pp 115 โ 139. British Grassland Society,
Hurley.
DairyMod and the SGS Pasture Model documentation 66
Sands PJ (1995). Modelling canopy production. I. Optimal distribution of photosynthetic resources,
Australian Journal of Plant Physiology, 22, 593-601.
Thornley JHM (1970). Respiration, growth and maintenance in plants. Nature 227, 304 โ 305.
Thornley, JHM (1998). Grassland Dynamics, An Ecosystem Simulation Model. CAB International,
Wallingford, UK.
Thornley JHM and Johnson IR (2000). Plant and Crop Modelling. Reprint of 1990 Oxford University Press
edition. www.blackburnpress.com.
Thornley JHM and France J (2007). Mathematical Models in Agriculture. CABI, Oxford.
Chapter 4: Water dynamics 67
4 Water dynamics
4.1 Introduction
The water balance in pasture and crop systems involves an intricate interaction between rainfall,
evapotranspiration, runoff and infiltration. The general scheme is presented in Fig. 4.1:
Figure 4.1: Schematic representation of the soil water balance.
In this diagram, the processes that are included in the model are shown. Some of these are quite readily
measured, whereas others are more difficult. For example, while rainfall can be measured quite accurately,
it is much more difficult to measure actual evapotranspiration.
The individual processes are now considered in turn.
4.2 Potential transpiration
The theory of canopy transpiration has received much attention, and the most widely used model is the
Penman-Monteith (PM) equation (Penman (1948) and Monteith (1965)). This approach is based on sound
physical principles, and describes the influence of radiation, temperature, vapour deficit, windspeed and
canopy structure on water use. As with any theory, there is always scope to incorporate greater
complexity, but the PM equation provides an ideal description of canopy water use for most crop and
pasture physiological studies, and is widely used in crop and pasture models. For further discussion see
Monteith (1973), or the later edition Monteith and Unsworth (2008), Campbell (1977), or the later edition
Campbell and Norman (1998), Jones (1992), Allen et al. (1998), Thornley and Johnson (2000), Thornley and
France (2007), Johnson (2013). Background definitions for water vapour and conductance are given in
Chapter 1, while the canopy radiation balance was discussed in Chapter 2: this material will be referred to
frequently throughout this Chapter. Canopy transpiration is influenced by canopy temperature which, in
turn, is affected by the prevailing environmental conditions. A key part of the analysis for canopy
transpiration is the elimination of canopy temperature so that transpiration is defined in terms of air
temperature and other environmental factors. For a complete derivation of the PM equation as used here,
soil water
movement
rain
irrigation
ET
run-off
through drainage
transpiration
canopy litter soil
evaporation
DairyMod and the SGS Pasture Model documentation 68
see Thornley and Johnson (2000) or Johnson (2013). The approach here is to define potential transpiration
and use this in conjunction with available soil water to estimate the actual transpiration by the plant
canopy, as discussed in Chapter 3, Section 3.2.
For daily crop and pasture models that work with standard meteorological data, transpiration is generally
required in units of mm d-1 to be consistent with rainfall. However, transpiration is also presented in mol
units in the literature. The conversion is straightforward with
1 mol water โก 0.018 kg water (4.1)
and
1 kg water m-2 โก 1 mm water (4.2)
so that
1 mol water m-2 โก 0.018 mm water (4.3)
The PM equation involves defining conductances for water vapour which can be prescribed either as m s-1
or mol m-2 s-1. The present theory is presented using mass rather than mol units.
4.2.1 Penman-Monteith equation
The PM equation defines the canopy transpiration rate in terms of climatic conditions and canopy
parameters. It can also be adapted to define evaporation from soil, litter or the canopy surface. The
general approach in deriving and using the PM equation is to start with the instantaneous rate of
transpiration (mm s-1) and then scale that up to get daily values. The general formulation for the PM
equation for daily transpiration, assumed to occur during daylight hours, can be written as
86 400
1
, ,,n day day p a a v aT
a c
sR f c g eE
s g g
(4.4)
where the symbols (some of which have been defined in Chapter 2), with units, are defined by:
๐ธ๐ Daily transpiration rate mm water d-1
๐ ๐,๐๐๐ฆ Canopy net radiation balance J m-2 d-1
๐๐๐๐ฆ Daylight fraction -
ฮ๐๐ฃ,๐ Vapour pressure deficit Pa
๐ Latent heat of vaporization 2.45 x 106 J kg-1
๐ Slope of the saturated vapour pressure as a
function of temperature relative to ๐
Pa K-1
๐พ Psychrometric parameter Pa K-1
๐๐ Specific heat capacity of dry air J kg-1 K-1
๐๐ Density of dry air 1.2 kg m-3
๐๐ canopy conductance m s-1
๐๐ boundary layer conductance m s-1
Chapter 4: Water dynamics 69
Vapour pressure deficit was discussed in Chapter 1, Section 1.6.2; the slope of the saturated vapour
pressure as a function of temperature, ๐ , is calculated by differentiating the Tetens formula, eqn (1.97) in
Chapter 1, and is
6
2
2 58 10 17 5
241241
. .exp
Ts
TT
(4.5)
Density of dry air is related to temperature according to
352.9
273.2T
(4.6)
The psychrometric parameter, ๐พ, is given by
p atmc P
(4.7)
where ๐๐๐ก๐ is atmospheric pressure, taken to be
0 1
11000
,.
atm atm seaP P
(4.8)
where ฮ, m, is the altitude and the atmospheric pressure at sea level is
101 325, PaatmP (4.9)
so that ๐๐๐ก๐ falls by 10% for every 1000m increase in altitude. The final parameter in eqn (4.7) is ํ ,the
ratio of the relative molecular mass of water to that of dry air and is (eg, Thornley and Johnson, 2000), with
value
0 622. (4.10)
It now remains to define the canopy and boundary layer conductances, which are the conductances of
water vapour across the leaf stomata and from the surface of the leaves to the bulk air stream respectively.
Canopy conductance
Canopy conductance is generally defined as the product of the leaf stomatal conductance and the
conducting leaf area of the canopy. In the present model, there is no attempt to simulate leaf stomatal
conductance and so canopy conductance is taken to be a constant value appropriate for well watered
leaves. This means that eqn (4.4) applies to plants that are not under water stress. This is then used in eqn
(3.5) in Chapter 3 where plant response to transpiration demand is discussed. The constant values for
stomatal conductance are:
1
1
0 015
0 01
3
4
C : . m s
C : . m s
c
c
g
g
(4.11)
where the C3 value is similar to that used by Allen et al. (1998) and the lower C4 value is because these
plants have a lower internal leaf CO2 concentration due to the C4 photosynthetic mechanism which results
in a lower stomatal conductance.
Boundary layer conductance
There is considerable discussion of boundary layer conductance in the literature, although not without
problems, as discussed in Johnson (2013). Rather than attempt to add greater complexity to the theory, a
DairyMod and the SGS Pasture Model documentation 70
much simpler approach is used here to capture the general characteristics of the expected behaviour for
canopy conductance. ๐๐ is assumed to be related to mean daily windspeed, ๐ข (m s-1), and canopy height, โ
(m), given by
0 5
0 0
.
, , ,a a a ref aref ref
u hg g g g
u h
(4.12)
where ๐๐,๐๐๐ is the value of ๐๐ at the reference windspeed ๐ข๐๐๐ and canopy height โ๐๐๐. The reference
values for windspeed and canopy height are taken to be
12
0 3
m s
. m
ref
ref
u
h
(4.13)
and the default conductance parameters are
10
1
0 002
0 01
,
,
. m s
. m s
a
a ref
g
g
(4.14)
These values are consistent with Allen et al. (1998).
4.3 Potential daily evaporation
Potential evaporation of water sitting on the surface of the leaves, litter or soil is treated in a similar fashion
to transpiration but now there is no resistance to water movement through the leaf stomata, so that the
term 1 ๐๐โ โ 0 in eqn (4.4) and hence
86 400, ,,n day day p a a v av
sR f c g eE
s
(4.15)
Although there may be some nighttime evaporation, the reverse may also occur, with deposition of dew.
Neither of these are likely to be significant and are not considered in the model.
4.4 Soil water infiltration and redistribution
The description of soil water infiltration and redistribution is crucial in the study of water movement in
soils. The aim of this section is to consider some approaches and issues relating to this topic. There is a
large body of theory on modelling soil water dynamics, and so the objective here is to give a brief account
of some of the principal approaches and then discuss the particular options that are available in the present
model.
The most widely used mechanistic model for soil water infiltration is the Richards equation, which
combines Darcyโs law for water movement along a water potential gradient with mass balance. There is
considerable appeal to the Richards equation, but it does pose significant challenges in solving it for soils
with variable soil hydraulic properties. While earlier versions of the SGS Pasture Model and DairyMod had
the option of using the Richards equation, we have now moved to the more flexible and robust
โcapacitanceโ model, and so the Richards equation is not discussed here.
The flux of water, ๐ m water d-1, is given by
satsat
q K
(4.16)
Chapter 4: Water dynamics 71
where ๐ m3 water (m3 soil)-1 is volumetric soil water content, ๐๐ ๐๐ก saturated water content, ๐พ๐ ๐๐ก m d-1 is the
saturated hydraulic conductivity which is the value of ๐ when ๐ = ๐๐ ๐๐ก, and ๐ is a flux coefficient which is
discussed below.
๐๐ ๐๐ก is calculated from soil bulk density, ๐๐ kg (m3 soil)-1, and soil particle bulk density, ๐๐ kg m-3, where
1 bsat
p
(4.17)
which simply assumes that when the soil is saturated all pore spaces are occupied by water. The standard
default value
32 650, kg mp (4.18)
is used.
The โfield capacityโ, ๐๐๐, sometimes referred to as โdrained upper limitโ is defined here such that
fc fcq q (4.19)
where ๐๐๐ is a very small flow rate, taken to be
10 1. mm dfcq (4.20)
parameter ๐ is then derived from eqn (4.16) as
ln
ln
fc sat
fc sat
q K
(4.21)
The model has low, medium and high textured soils, as well as a duplex, as defaults. Default values for the
medium textured soil are:
1
3
0 15
1 2
0 4
. m d
. g cm
.
sat
b
fc
K
(4.22)
which, with eqns (4.17) and (4.21) give
0 55
23 3
.
.sat
(4.23)
These values are used to illustrate the flux, ๐, in Fig. 4.2. According to this approach:
only water in excess of the drainage point can move, and all movement is downwards;
the flux decreases as the available water for movement declines, as controlled by ๐, which in turn is
derived from the water holding capacity of the soil.
DairyMod and the SGS Pasture Model documentation 72
Figure 4.2: Water flux, ๐, eqn (4.16), in relation to soil water content for the parameters in
eqns (4.22) and (4.23)
The method is applied by divide the soil into layers โ in the present model, the first four are each 5cm and
then subsequent layers are 10cm. A sub-daily time-step is then calculated to ensure the water movement
from any layer does not exceed that which is available. This requires evaluating the variable
,
, ,
sat
sat fc
K
z
(4.24)
where โ refers to the soil layer and ๐ฟ๐งโ, m, is the layer thickness. The number of time increments in the
day is then chosen to ensure that it is greater than all of the ๐โ values. To avoid any likelihood of problems,
the number of time-steps is then increased by 50%.
A further constraint applies in selecting the daily time increments in that it must be an exact multiple of the
sub-daily time increment for the prescription of daily rainfall distribution as discussed in Chapter 2, Section
2.2. By default, rainfall is divided into hourly distributions and so the daily time increments for infiltration
must be a multiple of 24.
This model is readily implemented numerically and is formulated in terms of easily characterised soil
parameters. On the interface where the parameters are prescribed, the wilting point and air-dry water
content are also included. These are used for plant water use and soil water evaporation respectively and
not directly for infiltration.
The main difference between this model and the more commonly used tipping-bucket model is the use of
hydraulic conductivity and a fairly fine depth layer distribution. In the tipping-bucket model, the layers are
generally coarser and water in excess of field capacity is assumed to move from one layer to the next in a
day (although variations on this are to be found).
4.5 Runoff
There are several approaches in the literature for the treatment of runoff, depending on the general
objectives of the model as well as the spatial- and time-scales. For the present purposes, it is necessary to
be able to calculate the flux of water off the paddock on a time-scale that is consistent with the treatment
of rainfall inputs, infiltration and evapotranspiration, which are calculated at sub-daily intervals.
Two widely used, and quite similar, approaches are the Manning equation and Hortonโs equation.
Manningโs equation states that the speed of water movement across the surface, v (m s-1)
0.00 0.10 0.20 0.30 0.40 0.50
Soil water content, %vol
-2.0
-1.0
0.0
1.0L
og
[ flu
x (
cm
/ d
ay )
]
0.00 0.10 0.20 0.30 0.40 0.50
Soil water content, %vol
0
5
10
15
Flu
x, cm
/ d
ay
Chapter 4: Water dynamics 73
2 3 1 2D S
vn
(4.25)
where ๐ท (m) is the depth of water on the surface, ๐ is the profile slope (%), and ๐ (s m-1/3) is the โManning
coefficientโ, with typical values of around 0.04 for pastures and 0.1 for bare soil. Hortonโs equation is
similar in structure to eqn (5.18a). The main disadvantage in using eqn (4.25) is that ๐ depends on the
ground cover characteristics which will vary throughout the simulation and so a simplified approach is used.
While runoff will clearly depend on the depth of water on the soil surface, it will also depend on the ground
cover, and so it is assumed that:
1 20v D D S (4.26)
where ๐ท (m) is the depth of water on the surface, ๐ท0 is the surface detention (amount of water that the
bare soil surface can hold with no runoff), and ๐ (s-1) is related to the relative ground cover (from 0 to 1) by
0 0 covermx (4.27)
This approach captures the essence of runoff in that it increases with profile slope and depth of water on
the surface (above a threshold value), while increasing ground cover will decrease runoff. The parameters
๐0 and ๐๐๐ฅ can be prescribed on the model interface. Note that the ground cover components are derived
in Chapter 3, Section 3.3.
The runoff speed, as given by eqns (4.26), (4.27) is shown in Fig. 4.3.
Figure 4.3: Runoff speed as a function of excess surface water for a range of ground covers as
shown. The parameters are: ๐ = 5%, ๐0 = 100 100 s-1 and ๐๐๐ฅ = 10 10 s-1.
4.6 Evaporation
Water can evaporate from the canopy, litter or soil. Evaporation from the canopy occurs if there is any free
standing water on the canopy. The potential evaporation is defined by eqn (4.15). Water can evaporate
from the canopy, litter and soil. These are considered in turn
4.6.1 Canopy
It is assumed that any water sitting on the canopy surface (leaves) is available for evaporation at the
potential rate. If the amount of water on the canopy surface is ๐ป๐๐๐๐๐๐ฆ, then the actual evaporation of
water from the canopy is simply:
, max ,v canopy canopy g vE H f E (4.28)
Excess water (mm)
Ru
no
ff sp
ee
d (
cm/s
)
0
25%
50%
75%
100%
DairyMod and the SGS Pasture Model documentation 74
where ๐๐ is the total ground cover by the canopy, eqn (2.23) in Chapter 2, and ๐ธ๐ฃ is the potential daily
evaporation from a free surface, as given by eqn (4.15).
4.6.2 Litter
As for the canopy, it is assumed that if there is any water held in the litter then it is available for
evaporation. It is therefore necessary to define the ground cover due to the presence of litter, ๐๐๐๐ก๐ก๐๐ kg
d.wt m-2, which is defined as
0 69,,
exp . litterg litter
litter h
Wf
W
(4.29)
where the -0.69 factor is ln(2) and ensures that
0 5, , .g litter litter litter hf W W (4.30)
However, the evaporative demand is attenuated in relation to canopy cover. Thus, if the amount of water
held by the litter is ๐ป๐๐๐ก๐ก๐๐, then the actual evaporation of water from the litter is
, ,max , 1v litter litter g g litter vE H f f E (4.31)
Where, as above, ๐๐ is the total ground cover by the canopy, eqn (2.23) in Chapter 2, and ๐ธ๐ฃ is the potential
daily evaporation from a free surface, as given by eqn (4.15).
4.6.3 Soil
Soil evaporation is the flux of water from the soil to the atmosphere in response to evaporative demand,
ground cover and soil water content. It is assumed that the potential to evaporate from the soil declines
with depth according to the function
0 69, ,
, ,
min ,exp .
fc ad
h fc ad
z
z
(4.32)
where ๐งโ (m) is the mid-point of the depth of layer โ, the potential declines by 50% at ๐งโ, ๐โ is water
content in the layer, ๐๐๐ as discussed above is field capacity, ๐๐๐ is the air-dry water content so that soil
water content cannot fall below this value, and ๐ is the coefficient used in the infiltration characteristics
above, eqn (4.21).
The water available for evaporation in each layer is given by
, ,evap fcH z (4.33)
Potential evaporation is then defined as
1 1, ,v soil g g litter vE f f E (4.34)
The model then works through the layers starting at the top and removes water from each layer according
to eqn (4.33) up to the potential limit given by (4.34).
This empirical approach captures the characteristics of soil evaporation restricted by the canopy and litter,
as well as the reduction due to available soil water and depth of water in the soil.
Chapter 4: Water dynamics 75
4.7 Concluding remarks
A detailed account of the water dynamics has been presented. Evapotranspiration is based on the widely
used Penman-Monteith equation and the details of prescribing the net radiation balance as well as canopy
and boundary layer conductance have been discussed. A versatile treatment of soil water infiltration has
been described that uses readily available soil hydraulic parameters. Likewise, the description of runoff is
straightforward to parameterize.
Water balance is an interaction between a range of complex flows. The model structure should allow the
user to explore these fluxes and so gain understanding into the underlying behaviour of the system.
4.8 References
Allen RG, Pereira LS, Raes D, and Smith M (1998). FAO irrigation and drainage paper no. 56: crop
evapotranspiration. http://www.kimberly.uidaho.edu/ref-et/fao56.pdf
Campbell GS (1977). An introduction to environmental biophysics. Springer-Verlag, New York, USA.
Campbell GS and Norman JM (1998). An Introduction to environmental biophysics, second edition.
Springer, New York, USA.
Johnson IR (2013). PlantMod: exploring the physiology of plant canopies. IMJ Software, Dorrigo, NSW,
Australia.
Jones HG (1992). Plants and microclimate. Cambridge University Press, Cambridge, UK.
Monteith JL (1965). Evaporation and environment. Symposium of the Society for Experimental Biology, 19,
205-234.
Monteith JL (1973). Principles of environmental physics, Edward Arnold, London.
Monteith JL & Unsworth MH (2008). Principles of environmental physics, third edition. Elsevier, Oxford,
UK.
Penman HL (1948). Natural evaporation from open water, bare soil and grass. Proceedings of the Royal
Society of London, Series A, 193, 120-145.
Thornley JHM and France J (2007). Mathematical models in agriculture. CAB International, Wallingford, UK.
Thornley JHM and Johnson IR (2000). Plant and Crop Modelling. Reprint of 1990 Oxford University Press
edition. www.blackburnpress.com.
DairyMod and the SGS Pasture Model documentation 76
5 Soil organic matter and nitrogen dynamics
5.1 Introduction
The soil nutrient dynamics component of this model includes organic matter turnover and inorganic
nitrogen (N) mineralization or immobilization, movement in the soil (leaching), adsorption in the soil, and
atmospheric losses. The turnover of organic matter (OM) is important both for the carbon balance of the
system and also the mineralization and immobilization of inorganic N. The model requires the initial
organic and inorganic status to be defined in order to start the simulation, which are defined graphically on
the interface. For inorganic NO3 that is subject to leaching it is common to find a bulge somewhere down
the profile where N can accumulate through leaching. The interface allows the user to construct such a
bulge for the initial status. The supply of organic matter is from litter (dead plant material), dung and dead
roots. There are three soil organic matter pools (in addition to surface litter, dung and live roots): fast and
slow turnover, and inert. The inert material does not decay but must be accounted for as it will show up in
experimental measurements. The only source of organic matter to the inert pool is through fire. The
organic matter and inorganic N dynamics of the model are illustrated schematically in Fig. 5.1. The model
described here is relatively simple in structure compared with many other soil organic models, although it
does capture the general processes involved.
Since measurements of soil nutrients are made much less frequently than those for soil water, care must be
taken in data analysis. For example, the nitrification of ammonium (that is, the transformation from NH4 to
NO3) is affected by water status and temperature. Since most organic matter is near the surface, and since
organic matter breakdown involves the production of NH4 which is then transformed to NO3, the time at
which these components are measured in relation to climatic conditions will be important. This is
compounded by the fact that NO3 leaches freely with water movement, whereas there is very little
movement of NH4. Furthermore, both volatilization and denitrification (atmospheric losses of NH4 and NO3
respectively) are very episodic and so are extremely difficult to measure.
In the analysis, all pool dynamics are defined for the same layer distribution as is used in the soil water
dynamics. Organic matter is expressed as kg C m-3, with the associated N component as kg N m-3, and
inorganic NO3 and NH4 have units kg N m-3. However, on the model interface, percent is also used and, for
the inorganic N, mg N kg-1 which is equivalent to parts per million (ppm).
Chapter 5: Soil organic matter and nitrogen dynamics 77
Figure 5.1: Schematic representation of the organic matter and nitrogen dynamics.
5.2 Organic matter dynamics
5.2.1 Overview
Soil organic matter dynamics are generally modelled by using pools of organic matter with different
turnover rates. Early models of this type were developed by Van Veen & Paul (1981) and Van Veen et al.
(1984, 1985), McCaskill and Blair (1988), Parton et al. (1988). Since then, the multi-pool approach has been
extensively applied with well-known models being APSIM (Probert et al. 1998), RothC (Jenkinson 1990),
CENTURY (Parton et al. 1998), and SOCRATES (Grace et al., 2006). A fundamental challenge with soil
mineralization
plant uptake
litter
dead roots
N fixation
Soil OM
fast, slow, inert
decay
Inorganic N
NH4+ NO3
โ
leaching
volatilization denitrification
fertilizer
urine
dung
ash (fire)
immobilisation
DairyMod and the SGS Pasture Model documentation 78
carbon models comprising several pools is that it is possible to get similar overall carbon dynamics with
different rates of input and turnover, and so we must continually assess all aspects of the soil carbon
dynamics in the model including the description of plant growth and senescence as it feeds into the soil
carbon.
The approach in the present model has been to simplify the description of soil organic matter dynamics to
include dynamic fast and slow turn-over pools, plus an inert component. The fast and slow pools are
sometimes referred to as particulate organic matter and humus soil carbon. The inert carbon pool, which is
essentially charcoal, is not subject to turnover. Keeping the model relatively simple avoids having to define
a large number of parameters that are likely to have strong interactions and are difficult to estimate. The
only parameters required are the decay rate constants for the fast and slow pools (proportion that decays
per unit time), their efficiency of decay (proportion of carbon respired during decay), and the transfer rate
from the fast to slow pool. The N concentration of the inputs are also required, and are calculated
dynamically in the model. Soil carbon dynamics are also affected by temperature and soil water status. Soil
carbon dynamics are driven by inputs from the plant material, and its digestibility.
5.2.2 Organic matter turnover
The model is illustrated in Fig. 5.2. There are two dynamic pools representing fast and slow turnover
carbon, ๐๐น,๐ ๐๐๐ and ๐๐,๐ ๐๐๐ kg C m-3, and a third inert pool which is primarily charcoal. Note that SI units are
used throughout the model, although results are converted to familiar units (such as t C ha-1 in the top
30cm soil). Inputs from dead plant material and dung are transferred to ๐๐น,๐ ๐๐๐ . This is subject to decay
and also transfer to ๐๐,๐ ๐๐๐, which also decays but at a slower rate. During decay, carbon is respired as CO2,
with the remainder going to the fast turnover pool. Note that restricting the analysis to these three pools is
consistent with current recommended measureable soil carbon pools (Skjemstad et al. 2004). Although the
model only considers two dynamic pools, the decay characteristics of ๐๐น,๐ ๐๐๐ are related to the digestibility
of the inputs so that litter and dead roots from less digestible pastures will decay at a slower rate than
more digestible inputs.
Figure 5.2. Overview of the soil carbon dynamics.
Fast
turnover
Slow
turnover
Mineralization or
immobilization
Respiration
Inert
Fire
Biochar
Respiration
Inputs
Chapter 5: Soil organic matter and nitrogen dynamics 79
The process of breakdown involves utilization of carbon to produce microbial biomass with an associated
respiratory loss. If the rate constant for pool decay is ๐, and the efficiency of breakdown ๐, then for every
kg of carbon in this pool that decays, the production of microbial biomass is ๐ kg with the remaining
(1 โ ๐) being lost to respiration, as illustrated in Fig. 5.3. This general structure is applied to both the fast
and slow turnover pools.
Figure 5.3: Schematic representation of OM breakdown. See text for details.
Denoting the carbon mass in the fast and slow turn-over pools by ๐๐น,๐ถ and ๐๐,๐ถ kg C m-3 respectively,
applying the general structure illustrated in Fig. 5.3, their dynamics are described by
1,
, , ,
d
d
F CC FS F C F F F C S S S C
WI k W k Y W Y k W
t (5.1)
,, ,
S CFS F C S S C
Wk W k W
t
d
d (5.2)
where ๐๐น and ๐๐ (d-1) are the decay rates for the fast and slow pools, ๐๐น๐ (d-1) is the transfer coefficient for
movement from the fast to slow pool, ๐๐น and ๐๐ are the dimensionless efficiencies of fast and slow organic
matter decay, and ๐ผ๐ถ (kg C m-3 d-1) is the rate of carbon input, and ๐ก (d) is time. The corresponding
respiration is
1 1, ,F F F C S S S CR Y k W Y k W (5.3)
Now consider the associated nitrogen dynamics. The decay of organic matter is assumed to be through
digestion by biomass. The biomass pool is not modelled explicitly, and is taken to be part of the fast pool.
Defining the N fraction of the biomass as ๐๐ต,๐, kg N (kg C)-1 which is taken to be a fixed quantity, and the
corresponding N fractions for the pools as ๐๐น,๐ and ๐๐,๐, which will be variables that depend on the inputs
and decay parameters, the nitrogen dynamics corresponding to eqns (5.1) and (5.2) are
, ,
, ,
,, ,
F N B N B NN FS F N F F N F S S
F N S N
W f fI k W k W
t f fY Y k
d1
d (5.4)
,, ,
d
d
S NFS F N S S N
Wk W k W
t (5.5)
The associated N mineralization rate, which is the flux of N from the soil organic matter into the ammonium
pool, is
, ,
, ,, ,
B N B NN F F N F S S N S
F N S N
f fM k W Y k W Y
f f
1 1 (5.6)
output, ๐๐๐ถ
๐๐๐๐ถ
(1 โ ๐)๐๐๐ถ
biomass
respiration
DairyMod and the SGS Pasture Model documentation 80
If this is negative then immobilization of inorganic nitrogen occurs and it is assumed that this nitrogen can
be supplied either from the NH4 or NO3 pools.
These relatively simple equations completely define the soil organic matter dynamics, including carbon
assimilation and respiration as well as nitrogen mineralization or immobilization. I have used nitrogen
fractions of organic matter and biomass rather than C:N ratios which are more common. The analysis is
clearer to work with using fractions, although the C:N ratio is the inverse of the N fraction. Thus, the
default value for ๐๐ต,๐ is taken to be 1/8 which is equivalent to a C:N ratio in biomass of 8. In the
simulations, results are shown as C:N ratios.
5.2.3 Effects of water and temperature
Organic matter dynamics are influenced by soil water status and temperature (eg Davidson et al., 2000).
The rate constants ๐๐น๐, ๐๐น, ๐๐ are defined by
H T refk k (5.7)
where ๐๐ป and ๐๐ are dimensionless water and temperature functions respectively, and ๐๐๐๐ is a reference
value for each of the rate constants defined at non-limiting soil water conditions and 20ยฐC. Estimating
these responses from experimental data is difficult owing to variation in the data.
It is assumed that soil biological processes are unrestricted by available water at water potentials greater
than -100kPa (RE White, personal communication). Using the widely used Campbell water retention
function (Campbell, 1974) to relate soil water content to potential, it is readily shown that for a wide range
of soil types, the soil water content corresponding to -100kPa occurs close to the average of field capacity
and wilting point. As in Chapter 4, which discusses water dynamics in detail, denote the volumetric soil
water content by ๐ m3 water (m3 soil)-1, with field capacity and wilting points ๐๐๐ and ๐๐ค respectively, so
that the soil water content at -100kPa, ๐100 can be approximated by
100 0 5. w fc (5.8)
A versatile equation for ๐๐ป which is based on the temperature functions discussed in Section 1.35 in
Chapter 1, which increases from zero to 1 over the soil water content 0 to ๐100 is
100100 100
1001
,
,
pqfc
H fc
(5.9)
where q is a curvature coefficient and
100
100
fcp q
(5.10)
The temperature function, ๐๐ is taken to be given by eqn (1.41) in Section 1.35 of Chapter 1, with
reference temperature 20โฐC.
๐๐ป and ๐๐ are illustrated in Fig. 5.4: the illustration for where ๐๐ป is for generic soil hydraulic properties
and that for where ๐๐ is for the model defaults parameter values.
Chapter 5: Soil organic matter and nitrogen dynamics 81
Figure 5.4: Soil water (left) and temperature (right) response functions ๐๐ป and ๐๐
respectively. ๐๐ป is shown for generic soil hydraulic properties, with ๐ = 3 while the
illustration for ๐๐ uses the model default values where both the reference and optimum
temperatures are 20โฐC and the curvature coefficient is 2.
5.2.4 Influence of inputs on organic matter dynamics
Now consider the influence of the quality of organic matter inputs through plant and root senescence on
organic matter dynamics. For each plant species, the digestibility of both the live and dead plant tissue is
prescribed. The value for the dead material is taken to influence the decay coefficient, ๐๐น, of the fast pool.
This is done on a pro-rata basis, so that the decay coefficient on day ๐ก is related to the value on day ๐ก-1 by
1, , , , , , ,in
F t F ref C in F t F C F C C inref
k k W k W W W
(5.11)
where ๐๐น,๐ถ is the initial mass of carbon in the fast pool, ฮ๐๐ถ,๐๐ is the carbon input with digestibility ๐ฟ๐๐,
๐ฟ๐๐๐ is a reference digestibility (taken to be 0.4), and ๐๐๐๐ is the reference decay rate for material with
digestibility ๐ฟ๐๐๐.
It is assumed that the decay rates for the fast and slow pools are independent of soil type, whereas the
transfer from the fast to slow pool is taken to be related to the soil clay fraction. Thus,
,FS FS refref
k k
(12)
where ๐พ is the clay fraction and ๐พ๐๐๐ is a reference value so that ๐๐น๐ = ๐๐น๐,๐๐๐ when ๐พ = ๐พ๐๐๐. By default,
๐พ๐๐๐=0.3. In the model, the actual clay fraction is defined in the water module.
This completely defines the soil organic matter dynamics including carbon accumulation and respiration, N
mineralization and immobilization, and the influence of soil water, temperature, and quality of inputs. The
decay rate of the fast turnover pool will decline with decreasing quality of organic matter inputs, as defined
by digestibility.
5.2.5 Half-life and mean-residence time
The above analysis is formulated in terms of decay rates, ๐ d-1, that is proportion per day, for the soil
organic matter pools. These decay rates are generally very small โ for example, the model defaults for the
fast and slow turnover pools are 10-3 and 4x10-5 d-1. However, these parameters do not lend themselves to
intuitive biophysical interpretation. For linear decay systems of the form used here, where the time course
of pool ๐ with decay coefficient ๐, with no inputs to the system, is
0.0
0.2
0.4
0.6
0.8
1.0
ฯH
ฮธw ๐fc ๐sat
0.0
0.2
0.4
0.6
0.8
1.0
0 10 20 30
ฯT
Temperature, ยฐC
DairyMod and the SGS Pasture Model documentation 82
d
d
WkW
t (5.13)
which has solution
0ktW W e (5.14)
where ๐0 is the initial value of ๐. This is, of course, exponential decay. The half-life, โ, is the time taken
for ๐ to reach half its initial value, and is simply given by
2 0 69ln .
hk k
(5.15)
โ has units that are the reciprocal of ๐ and so, in the present model, โ has units of d. For pools with slow
turnover, it is common to convert this to years: for example, the default value of 4x10-5 d-1 for the slow
turnover pool is equivalent to 47.5 years. The model interface presents the half-lives for the fast turnover
pool as days and slow turnover pool as years.
Mean residence time (MRT) is also used. This again is derived from exponential decay in refers to the mean
time that an element, or constituent, of the pool remains in the pool, and is given by
1
rk
(5.16)
so that
0 69.
hr (5.17)
Thus the terms are linearly related.
My preference is for half-life and this is used in the present model.
5.2.6 Initialization
The organic matter pools need to be initialized at the start of the simulation. There are three generic
default soil organic matter types referred to as โLow OMโ, โMedium OMโ and โHigh OMโ which refer to the
initial organic matter status. Initial values for the fast and slow turnover pools, as well as the inert pool, are
defined for the top and base of the soil profile, along with a curvature and depth for 50% decline. The
interface then displays the organic matter to 30 cm as t C ha-1 as well as a percentage value. The C:N ratio
of each pool is also defined, along with the value for the biomass. The default initial soil carbon distribution
for the โMedium OMโ default soil type is shown in Fig. 5.5
Chapter 5: Soil organic matter and nitrogen dynamics 83
Figure 5.5: Initial soil carbon distribution for the default โMedium OMโ soil type.
The โlabileโ pool is the fast turnover pool. Note that the inert carbon is not shown as it does
not affect carbon dynamics.
5.2.7 Illustration
System dynamics will, of course, depend on climate, plant growth and organic matter inputs to the soil, and
variation in soil water content. Also, since soil carbon dynamics generally have low decay rates the
influence of initial conditions, in particular the mass of soil organic matter in the soil, may influence the
simulations for many years or decades. To demonstrate the general behaviour of the model, a simple
simulation is available on the model interface for fixed soil water and temperature effects and constant
organic matter inputs. This simulation is illustrated for the default โMedium OMโ soil (see the program
interface), a combined impact of soil water and temperature given by
0 5.H T (5.18)
and a daily input of 15 kg dwt ha (10cm)-1 d-1 with C:N ratio 25 and digestibility 0.4. The soil carbon percent
and C:N ratio are shown in Fig. 5.6 where the simulation has been run for 3650 days (approximately 10
years)
Figure 5.6: Sample simulation for soil organic matter dynamics under constant conditions as
discussed in the text. Left: soil carbon percent of each pool as well as the total. Right: C:N
ratio of the total, fast (labile) and slow turnover pools. Note that the C:N ratio of the inert
pool, which has default value 30, is not shown.
0.0 1.0 2.0 3.0
Soil carbon, % per soil
2.0
1.5
1.0
0.5
0.0
De
pth
, m
Total
Labile
Slow
0 1000 2000 3000 4000
Days
0.0
1.0
2.0
3.0
4.0
5.0
Ca
rbo
n, %
Total
Labile
Slow
Inert
0 1000 2000 3000 4000
Days
0
5
10
15
C / N Total
Labile
Slow
DairyMod and the SGS Pasture Model documentation 84
5.2.8 Surface litter and dung
Surface litter and dung turnover dynamics are treated the same as the fast turnover soil organic matter
pool. As litter and dung decay, carbon is respired with that remaining being transferred to the fast
turnover pool in the surface soil layer. The scale function of water, ๐๐ป is also taken to be that of the
surface layer. Individual decay rate and efficiency parameters for litter and dung are defined on the model
interface.
There is also physical incorporation of litter and dung into the soil. Again, this follows linear dynamics with
incorporation rates being defined for both the litter and dung.
Surface litter is treated exactly the same as organic matter as described above, but with the following
considerations:
It is assumed that the turnover rate for litter is half that of soil organic matter. This is because of
lower microbial levels.
The water content in the surface soil layer is used in the expression for the effect of water status on
litter turnover โ that is ๐(๐) in the above analysis.
There is a physical transfer of litter from the soil surface to the soil.
The depth to which litter can be transported and the rate constant for this transfer (proportion per day) are
prescribed. Litter is then transferred evenly to this depth at this rate.
5.3 Inorganic nutrient dynamics
5.3.1 Overview
Plants acquire N from the inorganic NO3 and NH4 pools. The mineralization, and possibly immobilization,
through organic matter dynamics has been described above. The other processes in the model are inputs
from fertilizer or urine, nitrification of ammonium, gaseous losses of N through denitrification of nitrate
and volatilization of ammonium, leaching and plant uptake. N uptake by the plants is described in the
Chapter 3, Section 3.5.
5.3.2 Nitrogen inputs
Nitrogen inputs can occur from urine or fertilizer. Urea inputs from fertilizer or urine are assumed to be
hydrolyzed immediately and incorporated in the surface soil NH4 pool, although some details apply.
Fertilizer inputs of nitrate or ammonium are transferred directly to the relevant surface inorganic N pool.
The partitioning of nutrients between dung and urine may play an important role in nutrient dynamics and
the associated plant response, since urine returns are readily available whereas for dung the process of
organic matter decay delays the release. This partitioning is discussed in Chapter 6. Urine N inputs are
transferred directly to the soil NH4 pool. While nutrient dynamics in urine patches are likely to differ from
the bulk soil due to the greater concentrations of nutrient in the patches, no explicit treatment for urine
patch dynamics is considered here. For urine inputs, the user specifies a maximum depth and scale factor
to distribute nutrient inputs.
5.3.3 Nitrification of ammonium
Nitrification of ammonium, which is the conversion of NH4 to NO3 is described using a Michaelis-Menten
response to available soil ammonium, so that the rate of nitrification is
Chapter 5: Soil organic matter and nitrogen dynamics 85
4
4
4,
4mx NH H T C
NH
NHV
NH K
(5.19)
where [๐๐ป4] is the ammonium concentration in the soil, mg N kg-1, ๐๐๐ฅ,๐๐ป4 is the maximum rate of nitrate
production, mg N kg-1 day-1, ๐พ๐๐ป4 is the NH4 concentration for half maximal response to ammonium
concentration, the ๐ functions are the water and temperature responses that are discussed I section 5.2.3
above, and ๐พ๐ถ represents the effect of soil microbial mass as described below.
According to this approach, the nitrification rate is linear in response to available soil ammonium at low
concentrations and then curves to an asymptote as the concentration increases. It is apparent from (5.19)
that at low concentrations
4
4
4,mx NH
H T CNH
VNH
K (5.20)
The default parameters are
4, 1mx NHV mg N kg-1 day-1 and
4100NHK mg N kg-1 (5.21)
The response with no limitations due to temperature, water or carbon is shown in Fig. 5.7.
Figure 5.7: The rate of nitrification as a function of available soil ammonium for non-limiting
water, temperature and carbon conditions as defined in eqn (5.19) with parameters in eqn
(5.21) and ๐๐ป = ๐๐ = 1 โ see text for discussion.
Since there is no direct treatment of the soil microbial pool, it is assumed that the labile soil carbon (fast
turnover) reflects the level of microbes, so that
,
,, ,
f CC L
f C ref
W L
W
(5.22)
where ๐ฟ represents the soil layer, and [.] indicates mg kg-1. The reference soil carbon concentration in the
fast turnover pool is
15000 0 5, , mgkg . %f C refW (5.23)
0 100 200 300 400
mg N (NH4) / kg soil
0.00
0.20
0.40
0.60
0.80
1.00
( m
g N
/ k
g s
oil )
/ d
ay
Maximum nitrificaton rate
DairyMod and the SGS Pasture Model documentation 86
5.3.4 Denitrification of nitrate
Denitrification is the conversion of nitrate to nitrous oxide and nitrogen gas and, while the actual
denitrification losses may be relatively small in terms of the overall nitrogen dynamics in the system, the
fact that nitrous oxide is such a major greenhouse gas (the CO2 equivalent value is currently taken to be
310), care must be taken with these calculations. It is generally assumed that denitrification responds to
temperature in an analogous manner to nitrification but that, since it is an anaerobic process, it only occurs
in wet soils and increases towards saturation. Furthermore, as the soil gets wetter, there is a shift from
losses from N2O to N2. In addition, as this is a microbial process, it is necessary to include the effect of soil
microbial mass. The dynamics of denitrification are now considered.
Denitrification is described using Michaelis-Menten dynamics for the response to available nitrate. As for
nitrification, denitrification is also related to temperature and soil water, and can be written as:
3
3
3,
3
mx NO T C
NO
NOV f
NO K
(5.24)
where [๐๐3] is the concentration of NO3 in the soil layer, mg N kg-1, ๐๐๐ฅ,๐๐3 is the maximum rate of nitrate
production, mg N kg-1 day-1, ๐พ๐๐3 is the NO3 concentration for half maximal response to nitrate
concentration, ๐๐ is the temperature response function discussed in Section 5.2.3 above, ๐๐(๐) is the
water response function , and ๐พ๐ถ again represents the effect of soil microbial mass as described by eqn
(5.22). The default parameters are:
3
0 1, .mx NOV mg N kg-1 day-1 and 3
20NOK mg N kg-1 (5.25)
and the rate of denitrification with these parameters is shown in Fig. 5.8.
Figure 5.8: Rate of denitrification, eqn (5.24) with the parameters in eqn (5.25).
The effect of water is a bit more complex and is an area that can have important implications on the
calculations of denitrification. The following approach is designed to allow flexibility in exploring the effects
of soil water content on denitrification. Water filled pore space rather than volumetric soil water content is
used, which is defined as
sat
(5.26)
0 20 40 60 80
mg N (NO3) / kg soil
0.000
0.020
0.040
0.060
0.080
0.100
( m
g N
/ k
g s
oil )
/ d
ay
Maximum denitrificaton rate
Chapter 5: Soil organic matter and nitrogen dynamics 87
where (as usual) ๐ is the volumetric soil water content and ๐๐ ๐๐ก is the saturated water content. According
to this definition, ๐ ranges between 0 (no water in the soil, which is generally not possible) to 1 at
saturation. It is now assumed that the water function, ๐๐(๐), in eqn (5.24) is given by
2 1
0
,,
,
,
sin , ;
, .
qmn dn
mn dnmn dn
mn dn
f
(5.27)
where ๐๐ is a curvature coefficient. The default parameters are
0 6, .mn dn and 2.q (5.28)
The partitioning of denitrification between N2O and N2 is defined according to the scheme described by
Granli and Bockman (1994) according to which by assuming that:
initially as the soil wets up all losses are to N2O โ this occurs between ๐๐๐,๐๐ and ๐๐๐,๐2
as the soil gets wetter there is a linear shift towards N2 losses โ this occurs between ๐๐๐,๐2 and
๐๐๐,๐2๐
at water contents greater than ๐๐๐,๐2๐ all denitrification losses are as N2.
Granli and Bockman (1994) described these dynamics qualitatively and for the present model I have
adopted the following mathematical scheme, whereby the functions for partitioning total denitrification
๐๐ฃ(๐) into N2O and N2 components, ๐๐ฃ,๐2๐(๐) and ๐๐ฃ,๐2(๐) respectively, are given by
2
2
,
,
,
1 ,
v N O
v N
f f
f f
(5.29)
where
2
22 2
2 2
2
, ,
,, ,
, ,
,
1, ,
, ,
0, .
mn dn mn N
mx N Omn N mx N O
mx N O mn N
mx N O
(5.30)
The default parameters are
2
0 7, .mn N and 2
0 9, . ,mx N O (5.31)
with ๐๐๐,๐๐ prescribed in (5.28).
The full denitrification function, with partitioning between N2O and N2 is illustrated in Fig. 5.9
DairyMod and the SGS Pasture Model documentation 88
Figure 5.9: Total N denitrification function, including N2O and N2 components, as functions of
water filled pore space, with parameters given by eqns (5.28), (5.31).
It is interesting to note that with this treatment of denitrification, soils with field capacity close to
saturation may be susceptible to more denitrification than soils where there is quite a difference between
saturation and field capacity. For example, if the saturated water content is 55% and field capacity is 45%,
then field capacity occurs at a WFPS of 80% which means that denitrification occurs at field capacity and
below (down to WFPS of 60% with the defaults here). Alternatively, if the field capacity is 30%, then this
corresponds to a WFPS of 54%, and denitrification will not occur. This means that once the soils are wet,
those soils with field capacity greater than the cut-off WFPS for denitrification may have greater rates of
denitrification.
A characteristic of the mathematical treatment is that by changing the exponent ๐๐, not only does the
shape of the total denitrification curve change, but so does the partitioning. This is illustrated in Fig. 5.10
which shows the responses for ๐๐ = 1 and ๐๐ = 3.
Figure 5.10: Total N denitrification function, along with the N2O and N2 components,
corresponding to Fig. 5.9, but with ๐๐ = 1 (left) and 3 (right). See text for details.
The empirical approach used here for partitioning denitrification into N2O and N2 components captures the
general characteristics described by Granli and Bockman (1994) and is quite straightforward to implement
in terms of parameters that have direct biophysical interpretation.
0 20 40 60 80 100
WFPS, %
0
20
40
60
80
100
Sca
le fa
cto
r, %
Total
N2O
N2
Denitrification scale factor
0 20 40 60 80 100
WFPS, %
0
20
40
60
80
100
Sca
le fa
cto
r, %
Total
N2O
N2
Denitrification scale factor
0 20 40 60 80 100
WFPS, %
0
20
40
60
80
100
Sca
le fa
cto
r, %
Total
N2O
N2
Denitrification scale factor
Chapter 5: Soil organic matter and nitrogen dynamics 89
5.3.5 Volatilization of ammonium
Volatilization, the conversion of ammonium to ammonia gas, mainly occurs from urine patches, from urea
fertilizer shortly after application, and from the bulk soil ammonium pool in the top soil layer. Volatilization
is assumed to occur in relation to evapotranspiration, ๐ธ๐ธ๐ mm d-1, and the scale parameter
0,
max , ET
ET ref
E R
E
(5.32)
where ๐ mm d-1 is the daily rainfall and ๐ธ๐ธ๐,๐๐๐ is a reference evapotranspiration rate with default value
15, mm dET refE (5.33)
Volatilization from urea inputs either from fertilizer or urine, denoted by ฮฉ๐ข๐๐๐ kg N m-2 d-1 is assumed to
be given by
urea urea NU (5.34)
where ๐๐, kg N m-2 d-1, is the daily urea N input and ๐๐ข๐๐๐ is a dimensionless parameter that defines the
proportion of urea N lost as volatilization with default value.
0 2.urea (5.35)
According to this approach, urea N losses to volatilization are proportional to the inputs and
evapotranspiration in excess of rainfall.
Volatilization from the surface layer of the bulk soil NH4 pool is treated in a similar way and is taken to be
4 4 4 1,NH NH NHW (5.36)
where, ๐๐๐ป4,1 kg N m-2, is the inorganic NH4 in the surface soil layer and, again, ๐๐๐ป4 is a dimensionless
parameter with default value
4 0 01.NH (5.37)
which is equivalent to 1% per day.
5.3.6 Nutrient adsorption
Nutrient adsorption is a key component in the movement of nutrients through the profile: nitrate does not
adsorb and so is prone to leaching, whereas for many soil types most of the ammonium is adsorbed and so
is less likely to leach, although this may not be true for very sandy soils. Note that in this section, all
analysis uses SI units โ that is kg rather than mg โ although solute concentrations are generally discussed in
mg nutrient (kg soil)-1 or mg nutrient L-1 for solution in water. Mixing units can create problems with the
analysis and, while these are not insurmountable, it is better practice to use SI units and make conversions
at the end. The following analysis is general for any inorganic nutrient that is subject to adsorption and so
the analysis is described for any inorganic nutrient. Indeed, much of the work in this field has been done
for phosphorous.
Two commonly applied approaches for describing nutrient adsorption are to use either the Freundlich
equation or the Langmuir equation.
The Freundlich equation is a power law, as given by:
qa sC C (5.38)
DairyMod and the SGS Pasture Model documentation 90
where, for any nutrient ๐๐ข, ๐ถ๐ is the nutrient concentration in the soil, kg ๐๐ข (kg soil)-1, ๐ถ๐ is the nutrient
concentration in solution, kg ๐๐ข (kg water)-1 , ๐ and ๐ refer to adsorbed and solution respectively, and ๐
and ๐ are empirical parameters, with ๐ usually in the range 0.4 to 0.5 for a wide range of soils. Note that
the reciprocal of ๐ is sometimes used.
The Langmuir equation is a form of the rectangular hyperbola, and can be written as:
,
,
a mx sa
s a mx
C CC
C C
(5.39)
where ๐ผ [kg ๐๐ข (kg soil)-1] [kg ๐๐ข (kg water)-1]-1 is the initial slope of the curve at low values of ๐ถ๐ , and ๐ถ๐,๐๐ฅ
kg nu (kg soil)-1 is the maximum adsorption capacity of the soil at saturation. The parameters can be
expressed mathematically as:
a sC C as 0sC , and ,a a mxC C when s sC C . (5.40)
With either of these equations, the analysis relates adsorbed to solution nutrient concentrations. However,
since the model defines total inorganic nutrient mass, it is necessary to calculate each of these components
from the total. An advantage of the rectangular hyperbola is that it allows analytical solution of the
individual concentrations in terms of the total nutrient in the soil, whereas the use of the power law, eqn
(5.38), requires a numerical approach. A second advantage is that the linear characteristic of the
adsorption curve at low nutrient levels is generally more realistic. Figure 5.11 shows the rectangular
hyperbola eqn (5.39) fitted to the data of Moody and Bolland. (1999) for phosphorous adsorption in three
contrasting soils, and it can be seen that the curve gives a good fit to each data set.
Figure 5.11: Phosphorous sorption curves using eqn (5.39) along with data from
Moody and Bolland (1999) for a Ferrosol (red), Vertosol (green), Podosol (blue).
The parameters for the soil types in Fig. 5.7 are
๐ผ (soln / sorbed at low P) ๐ถ๐,๐๐ฅ (max P sorption capacity)
Ferrosol 1225 495
Vertosol 661 150
Podosol 270 123
0
50
100
150
200
250
300
350
400
450
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
P s
orb
ed (
mg P
/ k
g s
oil)
Solution P concentration (mg P / L)
Chapter 5: Soil organic matter and nitrogen dynamics 91
Equation (5.39) is used in the model to define the sorption characteristics of NH4 โ it is assumed that NO3
does not adsorb and is all in solution. The default values for NH4 adsorption are
1
1
1000
500,
adsorbed NH4 concentration solution NH4 concentration
mg NH4 kg soila mxC
(5.41)
The NH4 sorption curve with default parameters is shown for in Fig 5.12.
Figure 5.12: Solution NH4 as a function of sorbed NH4 using eqn (5.39) with default parameters
given by eqn (5.41).
The analysis for deriving the sorbed and solution nutrient components from the total nutrient content of
any soil layer is now presented. This analysis is presented for any generic nutrient, ๐๐ข, although only NH4 is
considered in the model at present. If the mass of nutrient in any layer of depth โ๐ง (m) is ๐ kg ๐๐ข m-2,
then the adsorbed and solution components are
a a bM C z (5.42)
and
s s wM C z (5.43)
where ๐๐, kg soil m-3 is the soil bulk density, ๐๐ค, kg water m-3 is the density of water, and ๐ (m3 water) (m3
soil)-1 is the volumetric soil water content, and
a sM M M (5.44)
is the total mass of nutrient in the layer.
It now remains to calculate the concentration components in terms of the total nutrient mass. Writing
m M z (5.45)
it is readily shown that
2 0, ,w s a mx b w s a mxC C m C mC (5.46)
which is a quadratic equation for ๐ถ๐ as a function of ๐. While this has two solutions, one of them is always
negative, and so the positive solution defines ๐ถ๐ : it is simple to calculate this solution.
In the model, plant uptake is described in terms of total inorganic nutrient and not the actual solution
component. While nutrient in solution is the source for the plant, there is a continual flux from sorbed sites
0.0 0.5 1.0 1.5 2.0
Solution, mg / kg
0
100
200
300
400S
orb
ed
, m
g / k
g
Ammonium
DairyMod and the SGS Pasture Model documentation 92
to the solution. It is likely that the roots are physically close to the soil particles so that the nutrient
concentration in the region near the roots may well differ from that in the bulk soil water. Furthermore,
for NH4 (highly adsorbing), the actual nutrient in solution at any particular time is only sufficient for a few
hours growth, so that there is a continual flux from sorbed to solution to the plant. The principal role of
nutrient adsorption in the model is in the description of nutrient leaching, with plant uptake being
described in relation to total nutrient. Consequently, N leaching will be dominated by the flux of NO3 which
is assumed not to adsorb.
5.3.7 Nutrient leaching
It is assumed that nutrients in solution can move with the water as the water moves through the soil
profile. Thus, in the in the soil water infiltration and redistribution dynamics discussed in Chapter 4, Section
4.4, the proportion of water moving from one layer to the next in each sub-daily time-step, ฮ๐ก d-1 is
calculated and the corresponding proportion of solution nutrient that moves is taken to be the same.
5.4 Concluding remarks
The treatment of soil nutrients has covered organic and inorganic dynamics, leaching, nitrogen
transformations, and gaseous nitrogen losses. The aim has been to avoid unnecessary complexity and yet
to encapsulate the key processes.
Care must be taken in analysing soil nutrient data since, the observed data for variables such as organic
carbon or inorganic nutrient concentration result from a series of fluxes. Since different combinations of
these fluxes can lead to similar system state variables, attention should focus on the fluxes and not just the
state variables. For example, if the rate of root growth and senescence is greater than actually occurs, but
the rate of organic matter turnover is also greater than occurs, then the actual soil organic matter values
from the model may agree well with observational data. Similarly, if the observations and data differ, then
this could be due either to errors in estimates of inputs to the pools or to fluxes out of the pools.
The study of soil nutrient dynamics is particularly complex since data are often hard to obtain due to the
slow turnover rates that are involved. The model provides a sound structure for analysing nutrient
dynamics, and for interpreting observational data.
5.5 References
Campbell GS (1974). A simple model for determining unsaturated conductivity from moisture retention
data. Soil Science, 117, 311 โ 314.
Davidson EA, Verchot LV, Cattanio, JH, Ackerman, IL, Carvalho JEM (2000). Effects of soil water content on
soil respiration in forests and cattle pastures of eastern Amazonia. Biogeochemistry 48, 53-69.
Grace PR, Jeffrey JN, Robertson GP, Gage SH (2006). SOCRATESโA simple model for predicting long-term
changes in soil organic carbon in terrestrial ecosystems. Soil Biology & Biochemistry 38, 1172โ1176.
Granli T and Bockman OC (1994). Nitrous oxide from agriculture.
Nor. J. Agric. Sci. (Suppl. 12): 128 Jones CA, Cole AN, Sharpley AN, and Williams JR (1984). A simplified soil
and plant phosphorus model: 1. Documentation. Soil Science Society of America Journal. 48, 800-805.
Karpinets TV, Greenwood DJ, and Ammons JT (2004). Predictive mechanistic model of soil phosphorus
dynamics with readily available inputs. Soil Science Society of America Journal. 68, 644-653.
McCaskill MR (1987). Modelling S, P and N Cycling in Grazed Pastures. PhD Thesis, University of New
England, Armidale, NSW, Australia.
Chapter 5: Soil organic matter and nitrogen dynamics 93
McCaskill MR, and Blair GJ (1988). Development of a simulation model of sulfur cycling in grazed pastures.
Biogeochemistry 5, 165-181.
Moody PW and Bolland MDA (1999). Phosphorus. In Soil analysis: an interpretation manual. (Eds KI
Peverill, LA Sparrow, DJ Reuter) pp. 187โ220. (CSIRO Publishing: Melbourne)
Parton WJ, Steward JBW, Cole CV (1988). Dynamics of C, N, P and S in grassland soils: a model.
Biogeochemistry 5, 109-131.
Parton WJ, Stewart JWB, Cole CV (1998). Dynamics of C, N, P and S in grassland soils: a model.
Biogeochemistry 5, 109-131.
Probert ME, Dimes JP, Keating BA, Dalal RC, Strong WM (1998). APSIMโs water and nitrogen modules and
simulation of the dynamics of water and nitrogen in fallow systems. Agricultural Systems 56(1), 1-28.
Skjemstad, JO, Spouncer, LR, Cowie, B, Swift, RS (2004) Calibration of the Rothamsted organic carbon
turnover model (RothC ver. 26.3), using measurable soil organic carbon pools. Australian Journal of Soil
Research 42, 79-88.
Thornley JHM (1998). Grassland Dynamics, An Ecosystem Simulation Model. CAB International,
Wallingford, UK.
Thornley JHM and Johnson IR (2000). Plant and Crop Modelling. www.blackburnpress.com.
Van Veen JA, Paul EA (1981). Organic carbon dynamics in grassland soils. 1. Background information and
computer simulation. Canadian Journal of Soil Science 61, 185-20.
Van Veen JA, Ladd JN, Frissel MJ (1984). Modelling C and N turnover through the microbial biomass in soil.
Plant and Soil 76, 257-74.
Van Veen JA, Ladd JN, Amato M (1985). Turnover of carbon and nitrogen through the microbial biomass in
a sandy loam and a clay soil incubated with C-14 glucose and N-15 ammonium sulfate under different
moisture regimes. Soil Biology and Biochemistry 17, 747-56.
DairyMod and the SGS Pasture Model documentation 94
6 Animal growth and metabolism
6.1 Introduction
Animal growth and metabolism is obviously a central component of DairyMod and the SGS Pasture Model.
Animal processes are modelled at different levels of complexity, ranging from detailed ruminant nutrition
models to simple growth curve response (for a discussion see Thornley and France, 2007). Detailed models
of rumen metabolism, while offering an understanding of processes such as animal response to feed
composition (e.g., Baldwin et al., 1987; Dijkstra et al., 1992; Dijkstra, 1994; Baldwin, 1995; Gerrits et al.,
1997; Thornley and France, 2007), may be too complex to be readily parameterized for different animal
types and breeds, or to apply routinely in biophysical pasture simulation models. Similarly, describing
animal growth directly with growth functions, such as the Gompertz equation, may give a reliable
description of experimental data, but this approach alone cannot be applied directly to conditions of
variable available pasture. For a whole-system biophysical model, striking a balance among complexity,
realism, and versatility allows the model to be applied quite readily to different animal breeds and respond
dynamically to pasture availability and quality.
The present model is based on the animal growth model described by Johnson et al. (2012) and the
development of that model to include pregnancy, lactation and nitrogen dynamics (Johnson et al., 2016). It
is an energy-driven model of animal growth and metabolism that has been developed specifically to be
appropriate for a whole-system biophysical pasture simulation model and includes responses to combined
pasture, concentrate, mixed ration and forage feed supply.
The model describes animal growth and energy dynamics for cattle and sheep in response to available
energy, and includes body protein, water, and fat. Model parameters have direct physiological
interpretation, which facilitates prescribing parameter values to represent different animal species and
breeds. Animal protein weight is taken to be the primary indicator of metabolic state, while fat is regarded
as a potential source of metabolic energy for physiological processes, such as the resynthesis of degraded
protein. The growth of protein is defined using a Gompertz equation, which is widely used in animal
modelling for sigmoidal growth responses, and was discussed in Section 1.3.6 in Chapter 1. Fat growth is
secondary and depends on current protein weight, as well as maximum potential fat fraction of body
weight (BW), which varies throughout the growth of the animal as defined by total BW. Protein is subject
to turnover. Therefore, maintaining current protein reserves requires the resynthesis of degraded proteins.
This maintenance, along with the energy required for activity, takes precedence over growth of new tissue.
New growth of fat depends on current protein weight, as well as the maximum potential fat fraction of BW,
with this maximum varying throughout the growth of the animal. While the Gompertz equation could also
be used to describe fat growth as done by Emmans (1997), the present approach allows the model to be
adapted to respond to restricted energy intake by viewing fat as a stored source of energy. Therefore,
body composition during growth and at maturity is determined by available energy with (as will be seen)
reduced fat fraction generally occurring when energy is restricted. The release of energy reserves from fat
during lactation is an important aspect of energy dynamics, particularly in dairy cows, and details are
presented describing the priority for milk production following parturition.
Animal intake in response to available pasture and pasture quality is described, as well as intake from
supplementary feeding in relation to quality. The composition of pasture, silage, hay or concentrate during
animal intake have a direct effect on animal growth and metabolism, including lactation, as well as nutrient
dynamics and the nitrogen contents of dung and urine. The model is formulated using standard
Chapter 6: Animal growth and metabolism 95
information on feed composition, and parameters relating to the energy dynamics in the animal, including
methane emissions during fermentation and the energy costs of dung and urine.
Energy dynamics in the animal are affected by the digestibility of the feed and also include costs for
excreted urine N and dung. The metabolic energy of feed is therefore also related to diet quality and N
composition and this is calculated in the model.
The model is described in complete detail here but, for further information and background, see Johnson et
al. (2012), Johnson et al. (2016).
6.2 Body composition during growth
Denoting empty BW by ๐ kg, and protein, water, and fat components by ๐๐, ๐๐ป, ๐๐น kg respectively,
these are related by
P H FW W W W (6.1)
The ash component of BW is not specifically included as it is generally a small proportion of the total and is
proportional to protein (Williams, 2005). It is assumed that water and protein weights are in direct
proportion, so that
H PW W (6.2)
where ๐ is a dimensionless constant. Thus, eqn (6.1) becomes
1 P FW W W (6.3)
The ratio of water to protein is assumed to be different for sheep and cattle (Johnson et al., 2012) and
default values are
3 5
3
.
sheep:
cattle: (6.4)
Protein is the primary component of growth with fat being related to protein. Defining body fat fraction as
FF
Wf
W (6.5)
kg fat (kg empty BW)-1, eqns (6.3) and (6.5) can be combined to give the individual protein, water, and fat
components as
1
1
1
1
;
;
.
FP
FH
F F
fW W
fW W
W f W
(6.6)
Body fat fraction is generally seen to increase with BW (e.g., Fox and Black, 1984; Lewis and Emmans,
2007). Normal growth conditions are defined as those under which intake is sufficient for potential protein
growth and associated fat growth, with the corresponding fat fraction at maturity denoted by ๐๐น,๐๐๐ก,๐๐๐๐.
It is assumed that during growth, the normal fat fraction increases linearly so that
,, , , , ,
,
F norm norm bF norm F b F mat norm F b
mat norm b
W W Wf f f f
W W W
(6.7)
DairyMod and the SGS Pasture Model documentation 96
where ๐๐, kg, is the birth weight, ๐๐น,๐ is the fat fraction at birth and subscripts ๐๐๐ก and ๐๐๐๐ refer to
โmatureโ and โnormalโ. Combining eqns (6.3) and (6.7) gives a quadratic equation for ๐๐น,๐๐๐๐ as a function
of ๐๐, which is
2 0, ,F norm F normaW bW c (6.8)
where the coefficients are
1 2 1
1 1 1
, , ,
,
,
,
F mat norm F b
mat norm b
F b P b
F b P P P b
f fa
W W
b f a W W
c f W aW W W
(6.9)
which is solved in the standard way, with the physiologically valid solution being
214
2,F normW b b ac
a
(6.10)
to give the normal fat weight, ๐๐น,๐๐๐๐, as a function of current protein weight, ๐๐, the birth fat fraction,
๐๐น,๐, and the normal mature fat fraction, ๐๐น,๐๐๐ก,๐๐๐๐.
For growth above normal, BW increases are entirely in the form of fat so that at maximum mature BW,
๐๐๐๐ก,๐๐๐ฅ, the protein weight is the same as that at normal mature BW, and hence
1 1, , , , , ,F mat norm mat norm F mat max mat maxf W f W (6.11)
where ๐๐น,๐๐๐ก,๐๐๐ฅ is the maximum fat fraction at maximum mature empty BW, ๐๐๐๐ก,๐๐๐ฅ, which gives
1
1
, ,, ,
, ,
F mat normmat max mat norm
F mat max
fW W
f
(6.12)
for ๐๐๐๐ก,๐๐๐ฅ in terms of the normal mature BW and corresponding prescribed fat fractions, so that
๐๐๐๐ก,๐๐๐ฅ is a derived quantity and not a prescribed parameter. With the default values for cattle and
sheep (see below), the mature maximum weight is 27% and 12% greater than the normal for cattle and
sheep respectively. Finally, during growth, the ratio of the maximum fat component of empty BW to that
during normal growth is taken to be constant, so that
1
1
, , , ,, ,
, , , ,
F mat max F mat normF max F norm
F mat norm F mat max
f fW W
f f
(6.13)
These equations completely define the normal and maximum empty BW during growth, as well as the
water and fat components, in terms of the fat fractions at birth, normal mature weight, and maximum
mature weight (๐๐น,๐, ๐๐น,๐๐๐ก,๐๐๐๐, ๐๐น,๐๐๐ก,๐๐๐ฅ , respectively) in terms of the current protein weight (๐๐) and
normal mature weight (๐๐๐๐ก,๐๐๐๐).
Default body fat composition parameters for sheep and cattle are:
0 06 0 3 0 45
0 1 0 25 0 33
, , , , ,
, , , , ,
. , . , .
. , . , .
F b F mat norm F mat max
F b F mat norm F mat max
f f f
f f f
cattle:
sheep: (6.14)
with units kg fat (kg BW)-1.
Chapter 6: Animal growth and metabolism 97
Values for birth weights (used below) and normal mature weights will vary between different animal types
and breeds, with the default values in the model being
50 600
6 60
,
,: ,
b mat norm
b mat norm
W W
W W
cattle: kg, kg
sheep kg kg (6.15)
6.3 Growth and energy dynamics
For potential protein growth, the net accumulation of protein, which includes protein synthesis and
degradation, is assumed to be defined by the Gompertz equation (see Section 1.3.6 in Chapter 1), which
can be written
DtPP
WW
t
de
d (6.16)
where ๐ก (d) is time, ๐ (d-1) is the initial specific growth rate for ๐๐, and ๐ท (d-1) is a parameter defining the
decay with time of the specific growth rate. This equation has solution
1
, exp
Dt
P P bW WD
e (6.17)
where ๐๐,๐ is the initial, or birth, protein mass. The mature, or asymptotic, protein weight is
/, ,
DP mat P bW W e (6.18)
so that
, ,ln P mat P b
DW W
(6.19)
Although eqn (6.17) is an analytical solution for ๐๐ through time for potential growth, the model needs to
address the situation where intake demand is not satisfied, and so it is convenient to write eqn(6.16) for
protein growth rate as a rate-state equation so that it is independent of time. This is readily derived by
eliminating the term ๐โ๐ท๐ก by using eqn (6.17) giving
,ln
P matP
P
WWDW
t W
d
d (6.20)
For more details, see Section 1.3.6 in Chapter 1. According to this formulation, the Gompertz equation for
๐๐ is written in terms of its final value, ๐๐,๐๐๐ก, and parameter ๐ท, eqn (6.19), which depends on the initial
value ๐๐,๐ and growth coefficient ๐. The default values for ๐ are
1
1
0 012
0 04
.
.
cattle: d
sheep: d (6.21)
with ๐๐,๐ evaluated from eqn (6.6).
In the following analysis, energy costs associated with growth are calculated according to the standard
approach whereby if the energy content of body tissue is ํ MJ kg-1 and the efficiency of growth is ๐, then
the energy required per unit growth, ๐ธ MJ kg-1, is
DairyMod and the SGS Pasture Model documentation 98
EY
(6.22)
The corresponding energy lost as heat during the synthesis of 1 kg due to respiration, ๐ MJ kg-1, is
1 Y
RY
(6.23)
where heat loss is accompanied by respiration of CO2. Default values for energy contents and efficiencies
for protein and fat synthesis are
1
1
23 6 0 48
39 3 0 71
. , .
. , .
P P
F F
Y
Y
protein: MJ kg
fat: MJ kg (6.24)
with the same values being used for sheep and cattle. See Johnson et al. (2012) for a discussion of the
derivation of these values. It can be seen that the energy costs of synthesising 1 kg of protein, excluding
the costs of resynthesis of degraded protein and fat are 49.2 and 55.4 MJ kg-1, respectively. However,
because protein growth also is associated with accumulation of body water (as discussed later), increasing
total BW by 1 kg with no actual fat growth requires substantially less energy. Therefore, it is important
when discussing the energy cost of growth to be clear as to the composition of the growth. As the animal
grows from birth to maturity, the fat composition generally increases and so the overall energy required
per unit of total BW gain will increase, as found by Wright and Russell (1984).
Once potential protein and fat growth are known, as well as energy costs for the resynthesis of degraded
protein and activity costs, the actual growth is calculated in relation to available energy intake. Under
restricted intake, fat catabolism may occur in order to supply energy for other processes.
Using eqn (6.22), the daily energy cost (MJ d-1) associated with protein growth as given by eqn (6.20) is
, ,P P
P g reqP
WE
Y t
d
d (6.25)
It is assumed that protein is subject to continual breakdown, with linear decay rate ๐๐ d-1, so that the
protein decay rate is
P Pk W (6.26)
and the energy required to resynthesis this protein (MJ d-1) is
1
, ,P maint req P P PP
E k WY
(6.27)
Default values for ๐๐ are
1
1
0 023
0 03
.
.
p
p
k
k
cattle: d
sheep: d (6.28)
Also, it is assumed that not all energy is released to the animal metabolic energy pool during protein decay,
but that some is lost as heat. Denoting this by ๐๐,๐, during protein decay the energy returned to the energy
pool is
, ,P d P d P P PE Y k W (6.29)
Chapter 6: Animal growth and metabolism 99
while the remainder of the energy is lost as heat, with the default value
0 9, .p dY (6.30)
Combining eqns (6.27) and (6.29), the net energy required for protein resynthesis (MJ d-1) is
1
, , , , , ,
,
P maint net req P maint req P d
P d P P PP
E E E
Y k WY
(6.31)
which is referred to as the protein maintenance energy requirement.
Now consider the growth of the fat component where it is assumed that
1,,
F FF g P
F max
W Wk W
t W
d
d (6.32)
where ๐๐น,๐, kg fat (kg protein)-1 d-1, is a fat growth parameter, with default values
1 1
1 1
0 03
0 2
,
,
.
.
F g
F g
k
k
cattle: kg fat kg protein d
sheep: kg fat kg protein d (6.33)
According to this equation, fat growth approaches the current potential maximum (๐๐น,๐๐๐ฅ) asymptotically,
with fat growth potential being directly related to current protein weight, ๐๐, so that absolute potential fat
growth increases as protein weight increases. Fat growth potential is related to protein weight because of
the assumption that metabolic state is defined by protein weight.
The energy required for fat growth, ๐ธ๐น,๐,๐๐๐ (MJ d-1), is, using eqn (6.22)
, ,F F
F g reqF
WE
Y t
d
d (6.34)
The final energy component to be included is that associated with animal physical activity, which is
assumed to be given by
act actE W (6.35)
(MJ d-1) where parameter ๐ผ๐๐๐ก MJ kg-1 d-1 is the energy requirement for animal activity per unit of BW.
Increasing this parameter will be appropriate, for example, for animals on hilly terrain. The default value
for both sheep and cattle is
1 10 025.act MJ kg d (6.36)
so that, for example, with a 600 kg steer ๐ธ๐๐๐ก=15 MJ d-1, whereas for a 60 kg sheep, it is 1.5 MJ d-1. While
other maintenance costs such as maintaining body temperature are not specifically included, these could
be included in this term if necessary.
Combining protein maintenance costs, eqn (6.31) with (6.35), gives the total maintenance energy
requirement as
, , , ,maint req P maint net req actE E E (6.37)
Equation (6.32) for the potential fat growth rate allows body fat to accumulate to the maximum. The
actual energy required to grow to normal fat weight during time โ๐ก (d) is simply
DairyMod and the SGS Pasture Model documentation 100
,
, , ,F norm FF
F g norm reqF
W WE
Y t
(6.38)
where, for a daily time-step model, โ๐ก=1 d. Of course, this will only be satisfied if
, , , , ,F g norm req F g reqE E (6.39)
Similarly, to grow to maximum fat weight, the energy required is
,
, , ,F max FF
F g max reqF
W WE
Y t
(6.40)
with
, , , , ,F g max req F g reqE E (6.41)
Finally, the energy required for normal growth is
, , , , , , ,req norm P g req maint req F g norm reqE E E E (6.42)
and for maximum growth
, , , , , , ,req max P g req maint req F g max reqE E E E (6.43)
6.4 Model solution in relation to available energy
The model described so far, describes growth rates for protein, the associated water and fat, as well as the
corresponding energy costs. In practice, growth is dictated by available energy and the theory is now
applied to this more general situation. Equations (6.25) and (6.34) define the energy required for
prescribed protein and fat growth rates, but they can be inverted to define growth rates in relation to
available energy, that is
, , , , , ,;P PP g avail P g avail P g req
P
W YE E E
t
d
d (6.44)
and
, , , , , ,;F FF g avail F g avail F g req
F
EW Y
E Et
d
d (6.45)
Forward differences with a daily time-step are used to calculate protein and fat components on day ๐ก (d) to
their values and growth rates on day ๐ก-1 according to
11
11
,, ,
,, ,
P tP t P t
F tF t F t
WW W t
t
WW W t
t
d
d
d
d
(6.46)
where โ๐ก is the time-step with
1t d (6.47)
Three sets of circumstances are now considered where the available intake energy, ๐ธ๐๐ (MJ d-1), exceeds
requirements for normal growth, is less than or equal to that for normal growth, but exceeds maintenance
requirements, or is less than maintenance requirements.
Chapter 6: Animal growth and metabolism 101
6.4.1 ๐ฌ๐๐ exceeds requirements for normal growth
If the available energy from intake, ๐ธ๐๐, exceeds requirements for normal growth then
, ,req norm in req maxE E E (6.48)
Protein growth and all maintenance costs are met, with any remaining energy being used for fat growth, so
that
, , ,
, ,
maint maint,req
P g P g req
F g in P g maint
E E
E E
E E E E
(6.49)
with ๐ธ๐,๐ and ๐ธ๐น,๐ being used in eqns (6.44) to (6.47) to calculate ๐๐,๐ก and ๐๐น,๐ก.
6.4.2 ๐ฌ๐๐ is between maintenance requirement and normal growth requirement
Under these circumstances
,maint in req normE E E (6.50)
and it is assumed that maintenance costs are met with the remainder of the available energy being fat and
growth, so that growth is reduced. The energy available for growth is partitioned between protein and fat
on a pro-rata basis according to requirement, so that
, ,, ,
, , , , ,
, ,, ,
, , , , ,
maint maint,req
P g reqP g in maint req
P g req F g norm req
P g reqF g in maint req
P g req F g norm req
E E
EE E E
E E
EE E E
E E
(6.51)
and, again, ๐ธ๐,๐ and ๐ธ๐น,๐ are used in eqns (6.44) to (6.47) to calculate ๐๐,๐ก and ๐๐น,๐ก.
6.4.3 ๐ฌ๐๐ is less than maintenance requirement
These animals do not have sufficient energy to grow, with all available energy being used for activity and
maintenance. For this scenario, ME intake is constrained by
,in maint reqE E (6.52)
and fat catabolism can occur.
As for fat growth, fat catabolism is assumed to be related to animal protein weight, which is an indication
of its metabolic state, and also related to available body fat. Therefore, it is assumed that the maximum
rate of fat catabolism is given by
,
, ,, ,
F F mnd mx F d P
F mx F mn
W WF k W
W W
(6.53)
(kg fat d-1) where ๐๐น,๐ [kg fat (kg protein)-1 d-1] is a fat decay parameter, so that the maximum daily rate of
fat catabolism is equivalent to the fraction ๐๐น,๐ of total protein weight at maximum fat composition.
During breakdown, there will be some energy lost as heat and so taking the efficiency of breakdown to be
๐๐น,๐, the ME available from fat catabolism is
DairyMod and the SGS Pasture Model documentation 102
, , , ,F d mx F d F d mxE Y F (6.54)
Default values for cattle and sheep are
1 10 005 0 95, ,. .F d F dk Y kg fat kg protein d (6.55)
The actual daily fat catabolism is now simply
, , , ,min ,F d F d mx m req inE E E E (6.56)
so that maintenance costs will be satisfied if there is sufficient energy available from fat catabolism,
otherwise the maximum energy available from fat will be utilized for partial satisfaction of maintenance
requirements.
According to this approach, if available energy from intake and fat catabolism does not meet maintenance
requirements there will be a reduction in protein weight and less activity. The reduction in activity is
consistent with reduced grazing. Note that fat catabolism does not occur to support new protein growth,
only the maintenance of existing protein.
6.5 Illustrations of animal growth dynamics
The following illustrations are presented to demonstrate the overall characteristics of the model as applied
to the growth of cattle and sheep. These illustrations are also discussed by Johnson et al. (2012).
The first set of illustrations consider growth and body composition for cattle and sheep under maximum
growth conditions, which allows us to compare the model results with observations summarised by Fox and
Black (1984) for cattle and Lewis and Emmans (2007) for sheep. In these papers the authors collated
experimental data and summarised relationships between body components with fitted empirical curves.
Summarising large amounts of experimental data in this way is one of the primary applications of empirical
models, as discussed by Thornley and France (2007). The default body composition and growth parameters
presented in the previous section have been selected based on these empirical responses, although the
specific mathematical formulation of those responses have not been used in the present model structure.
Figure 6.1 shows total empty BW growth, as well as protein, water, and fat components for sheep and
cattle for maximum growth. It should be noted that Fox and Black (1984) fitted polynomial curves for
protein, water, and fat as functions of total weight, whereas Lewis and Emmans (2007) related water and
fat to protein by using allometric equations. Consequently, the figures show the fitted curves for each body
component for cattle, but only water and fat for sheep. It can be seen from these figures that there is
virtually complete agreement between the present model and the curves that have been fitted to data, to
the extent that the dashed lines representing the data are largely obscured by the model responses. Apart
from this agreement, the general shapes of the responses are consistent with expected characteristics.
Chapter 6: Animal growth and metabolism 103
Figure 6.1: Empty BW and composition for cattle (left and sheep (right) from birth to maturity
for maximum growth. The solid lines are the model and the broken lines are the regression
curves derived by Fox and Black (1984) for cattle and Lewis and Emmans (2007) for sheep. Fox
and Black reported protein, water, and fat as functions of BW, while Lewis and Emmans gave
fat and water as functions of protein. Note that the model and observed response curves are
virtually identical and the response curves (broken lines) are almost completely obscured by
the model (solid lines).
The energy dynamics for cattle and sheep, corresponding to the growth characteristics in Fig. 6.1, are
illustrated in Fig. 6.2. It can be seen that energy requirement for protein growth peaks earlier than that for
fat growth, but as the requirements for protein growth decline the cost of protein maintenance increases
and reaches a greater value than the peak cost for new protein growth. In addition, maximum energy
requirement occurs prior to the animal reaching its maximum weight. Energy costs for the resynthesis of
degraded protein are considerably greater than activity costs, although this behaviour depends on the
choice of parameters for the protein degradation rate ๐๐ and activity costs, ๐ผ๐๐๐ก. One characteristic
difference apparent from Fig 6.2 is that the relative amount of energy required for maintenance is greater
in cattle than sheep.
0
100
200
300
400
500
600
700
800
900
0 500 1000 1500
We
igh
t, k
g
Time, days
0
10
20
30
40
50
60
70
80
0 100 200 300 400
Time, days
Total
Protein
Water
Fat
DairyMod and the SGS Pasture Model documentation 104
Figure 6.2: Top: growth energy dynamics for cattle (left) and sheep (right) from birth to
maturity, corresponding to Fig. 6.1. The total energy required, as well as the individual
requirements for protein growth, protein maintenance, fat growth, and activity are indicated.
Bottom: the combined growth and maintenance components are shown.
Note the different scales for cattle and sheep.
It is instructive to look at energy dynamics in relation to BW as well as through time. The responses for
growth, maintenance and total energy required, corresponding to Fig. 6.2, are shown in Fig. 6.3. There is a
non-linear relationship between the energy required for maintenance and total empty BW, which is often
characterised by an empirical allometric response. Although not shown here, this response is very similar
to the bodyweight raised to the power between 0.73 and 0.75, which is widely used in feed evaluation
systems and simulation models (ARC, 1981; Finlayson et al., 1995; National Research Council, 2001).
0
20
40
60
80
100
120
0 500 1000 1500
Ener
gy r
equ
irem
ent,
MJ
d-1
Time, days
0
2
4
6
8
10
12
14
16
18
0 100 200 300 400
Time, days
Total
P maint
P growth
Fat
Activity
0
20
40
60
80
100
120
0 500 1000 1500
Ener
gy r
equ
irem
ent,
MJ
d-1
Time, days
0
2
4
6
8
10
12
14
16
18
0 100 200 300 400
Time, days
Total
Growth
Maint
Chapter 6: Animal growth and metabolism 105
Figure 6.3: Total energy requirements, and the growth and maintenance components as
functions of empty body weight, for cattle (left) and sheep (right), corresponding to Fig. 6.2.
The analysis so far has considered growth under optimal conditions of non-limiting intake as defined by
๐ธ๐๐๐ , eqn (6.43) and now consider the situation where intake does not satisfy maximum demand. It may
be neither desirable nor practical for animals to grow to their absolute maximum, due to restricted feed or
the fact that maximum body fat may only be achieved through supplementary feeding. The illustrations in
Fig. 6.4 show animal growth with energy intake at maintenance plus 100%, 90%, 80%, 70% of potential
growth (protein and fat) energy requirement during animal growth, as given by eqn (6.42). The results are
as expected with growth being reduced under restricted intake. For example, the time to reach half
mature BW at full intake is 270 d for cattle and 70 d for sheep whereas with 70% intake requirement it is
342 d and 99 d, which correspond to increases of 27% and 41%, respectively.
Figure 6.4: Growth dynamics for growing cattle (left) and sheep (right) for intake either at full
requirement (100%) or maintenance plus 90%, 80%, and 70% growth requirement as
indicated. Note the different scales.
Animal growth rate and that of individual components varies through time and also in response to relative
intake. This is illustrated in Fig. 6.5 for both cattle and sheep, corresponding to the growth dynamics in Fig.
6.4 where the general pattern of the growth rate is consistent with sigmoidal growth. It can be seen that
0
20
40
60
80
100
120
0 200 400 600 800
Ener
gy r
equ
irem
ent,
MJ
d-1
Empty body weight, kg
0
2
4
6
8
10
12
14
16
18
0 10 20 30 40 50 60 70
Empty body weight, kg
Total
Growth
Maint
0
100
200
300
400
500
600
700
0 500 1000 1500
Emp
ty b
od
y w
eig
ht,
kg
Time, days
0
10
20
30
40
50
60
70
0 100 200 300 400
Emp
ty b
od
y w
eig
ht,
kg
Time, days
DairyMod and the SGS Pasture Model documentation 106
growth rates of all components are reduced as intake declines, and that the time for peak growth rate is
delayed, most noticeably for the fat component.
Figure 6.5: Total animal growth rate and that of the protein, water, and fat components of
body weight, as indicated, during growth for cattle (left) and sheep (right), corresponding to
Fig. 6.4. The solid lines are maintenance plus 100% growth requirement, large dashes 90%,
small dashes 80%, and dots 70%. Note the different scales.
The simulations in Figs 6.4 and 6.5 are for animals under feeding regimes that provide full maintenance plus
a fixed proportion of growth requirements. These illustrations are important as a means of examining the
modelโs performance but, in practice, the intake is likely to vary in response to both pasture quality and
availability, as well as management. The model can be applied directly to any feeding regime and can
respond to varying pasture availability. As a simple example, the above simulations are repeated but with
intake taken to be full maintenance plus a proportion of growth requirement that varies randomly between
70% and 100% of normal growth requirement, so that it fits somewhere between the illustrations shown in
Figs 6.4 and 6.5. This could apply, for example, to situations where supplementary feeding is provided to
ensure intake meets a required minimum. The results for empty BW, ๐, and energy requirements are
shown in Fig. 6.6 where it can be seen that, as expected, ๐ lies between the two fixed regimes. Also,
although there are fluctuations in energy supply, the actual growth curves for ๐ are quite smooth,
demonstrating that BW growth is buffered in relation to moderate fluctuations in intake.
One characteristic of the simulations illustrated in Figs 6.4, 6.5, and 6.6 for growing animals is that there
was no fat catabolism because, according to these feeding strategies, maintenance costs are always met.
In practice intake will vary and, particularly when animals are close to maturity, there may be some fat loss
to satisfy energy requirements. To explore this, the final set of illustrations considers mature animals with
intake reduced from mature maintenance requirement. The above analysis applies without modification,
although for animals at their mature optimum weight there will be no energy requirements for growth.
Consequently, for a mature animal that has less than its optimum protein or fat composition, intake
requirement may be greater than for the equivalent animal at optimum weight because there is a growth
energy requirement, notwithstanding the fact that activity costs will fall slightly as an animal loses BW. In
the following illustrations, that consider the effect of restricted intake on mature animals, intake is
prescribed as fractions of the mature maintenance requirement at optimum fat composition.
0
0.2
0.4
0.6
0.8
1
1.2
0 500 1000 1500
Gro
wth
rat
e, k
g d
-1
Time, days
0
0.1
0.2
0.3
0.4
0 100 200 300 400
Time, days
Total
Protein
Water
Fat
Chapter 6: Animal growth and metabolism 107
Figure 6.6: Top: growth dynamics for growing cattle (left) and sheep (right), for intake either
at full requirement (100%) or maintenance requirement plus 70% growth requirement, as well
as switching randomly between these two regimes, indicated by โRโ. Bottom: the
corresponding total, growth, and maintenance energy requirements, as indicated for the โRโ
simulations. Note the different scales.
The total empty BW, as well as the protein, water, and fat components, are shown in Fig. 6.7 for animals
receiving 90%, 80%, and 70% of mature maintenance requirement. It can be seen that in all cases the
weight components fall as expected. However, note that fat decline is virtually identical for the 80% and
70% regimes, which is due to fat catabolism occurring at the maximum rate, eqn (6.54). Consequently, the
protein weight decline is more rapid for the 70% regime. (The changes in protein weight may be difficult to
detect in this figure due to the relative size of this pool, although it should be noted that the fractional
decline in protein is identical to that for water since these components are in direct proportion, eqn (6.2).
0
100
200
300
400
500
600
700
0 500 1000 1500
Emp
ty b
od
y w
eig
ht,
kg
Time, days
0
10
20
30
40
50
60
70
0 100 200 300 400
Time, days
R
100
70
0
10
20
30
40
50
60
70
80
90
100
0 500 1000 1500
Ener
gy r
equ
irem
ents
, MJ
d-1
Time, days
0
2
4
6
8
10
12
14
16
0 100 200 300 400
Time, days
Total
Growth
Maint
DairyMod and the SGS Pasture Model documentation 108
Figure 6.7: Growth dynamics for mature cattle (left) and sheep (right) under a range of feed
intakes. Left: empty BW and components as indicated, with intake either at 90% (large
dashes), 80% (small dashes), or 70% (dots) of mature maintenance requirement as indicated.
(The colours and line styles are consistent with Fig. 6.5.)
The illustrations presented here demonstrate that the model gives realistic behaviour of cattle and sheep
growth under a range of energy intake levels.
6.6 Pregnancy and lactation
The analysis so far is for growing animals and pregnancy and lactation are now considered. The approach is
a natural extension of the growth model, and is described in detail in Johnson et al. (2016).
6.6.1 Pregnancy
Foetal growth is assumed to be exponential so that the growth rate is
f
f f
Wk W
d
dt (6.57)
where ๐๐, kg, is the foetus weight and ๐๐, d-1, is a growth coefficient. For normal growth, this equation is
solved to give
0,fk t
f fW W e (6.58)
If the pregnancy duration is denoted by ๐ก๐๐๐๐ then the foetus weight at ๐ก = ๐ก๐๐๐๐ is the birth weight, ๐๐,
and it follows that
0
,f pregk t
f bW W e (6.59)
Rather than prescribing the growth coefficient ๐๐, the parameter ๐๐ is defined as the fraction of the
pregnancy duration to 50% birth weight, so that
2 f f preg bW t t W (6.60)
which, with eqn (6.58), gives
0
100
200
300
400
500
600
700
0 100 200 300 400
Emp
ty b
od
y w
eig
ht,
kg
Time, days
0
10
20
30
40
50
60
70
0 100 200 300 400
Time, days
Total
Protein
Water
Fat
Chapter 6: Animal growth and metabolism 109
2
1
0 69
1
ln
.
fpreg f
preg f
kt
t
(6.61)
so that the foetal growth parameters to be defined are ๐ก๐๐๐๐ and ๐๐. Defaults are:
280 0 8
150 0 8
cattle: d
sheep: d,
, .
.
preg f
preg f
t
t (6.62)
Equation (6.58) with (6.59) and (6.61) defines foetal growth in terms of birth weight, pregnancy duration
and the time to 50% birth weight. ๐๐ is illustrated in Fig. 6.8 for a dairy cow โ the response for sheep has
the same characteristic shape but with different scale. It can be seen that, at conception, foetal weight
does not increase smoothly from zero โ the model could be refined to make this the case but the effect on
the simulations would be negligible and would not justify the extra complexity.
Figure 6.8: Foetal growth in dairy cows with default parameters.
If there are multiple foetuses, the birth weight is generally reduced so that, if the number of foetuses if ๐๐,
it is assumed that
2
1
, fb n bf
W Wn
(6.63)
where ๐๐ is the normal birth weight discussed in Section 6.2 above.
It can be seen that total foetal weight has been considered and not the individual protein, fat and water
components. This has been done to avoid unnecessary complexity. The energy requirement during
pregnancy is calculated in a similar manner to growth requirements, with the assumption that the fat and
protein composition during foetal growth is constant and the same as at birth. Thus, the energy required
for foetal growth is
,, ,b
, ,
P bF Ppreg req F preg f
F g b P g
WE f n
Y W Y (6.64)
where all terms have been defined previously with the exception of ๐พ๐๐๐๐ which is a scale factor to allow
for the fact that the energy required during pregnancy is greater than the energy density in the foetus. The
default value is
0 100 200 300 400
Days since parturition
0
10
20
30
40
50
Fo
etu
s w
eig
ht, k
g
DairyMod and the SGS Pasture Model documentation 110
2 preg (6.65)
which means that the energy requirement to grow 1 kg of foetus is double that of animal growth with the
same body composition (Rattray et al., 1974).
6.6.2 Lactation
The primary focus for the treatment of lactation is for dairy systems, although it is obviously important in
livestock systems with calves or lambs. Milk production can be defined either as L d-1 or kg milk solids d-1,
although milk solids are probably only relevant to dairy systems. Denoting the fat, protein and lactose
fractions of milk as ๐๐น,๐, ๐๐,๐, ๐๐ฟ,๐ respectively, and the density of milk as ๐๐, the fraction of milk solids,
๐๐,๐ ๐๐๐๐๐ kg solids L-1 is
, , ,M solids M F M P Mf f f (6.66)
and energy density of milk, ํ๐ MJ L-1 is
, , ,M M F F M P P M L L Mf f f (6.67)
where the energy densities for fat and protein were defined earlier, with values in (6.24), and the energy
density of lactose has default value
116 5.L MJ kg (6.68)
The efficiency of milk production, ๐๐, has default value
0 72 .MY (6.69)
so that the energy required for milk production, again applying eqn (6.22), is
1 MJ LMM
M
EY
(6.70)
Default values are
1
1
1 03 0 043 0 03 0 046
1 03 0 05 0 06
cattle: kg L
sheep: kg L =0.06,
, , ,
, , ,
. , . , . , .
. , . , .
M F M P M L M
M F M P M L M
f f f
f f f (6.71)
With these parameters, the net energy content of milk is 3.25 MJ L-1 and 4.66 MJ L-1 for cattle and ewes
respectively.
Actual milk production is defined in terms of available energy from intake and is discussed later.
Fat catabolism during lactation
The potential for fat catabolism to provide energy for other metabolic processes was discussed earlier.
During lactation, additional fat catabolism occurs to provide the extra energy requirements associated with
the production of milk. As time progresses, there is a shift from priority for milk production which incurs
fat catabolism, to replacing body fat through fat growth. This is defined in the model through the scale
function
0
0
,,
,
lact lact FF lact lact
lact lact F
(6.72)
Chapter 6: Animal growth and metabolism 111
where ๐๐๐๐๐ก is the time (d) since parturition. ๐๐น,๐๐๐๐ก lies between -1 and 1 as ๐ increases from zero, taking
the value 0 when ๐ = ๐๐๐๐๐ก,๐น0. The parameters are:
0
0
95
50
cattle: d
sheep: d
,
,
lact F
lact F
(6.73)
Equation (6.72) is illustrated in Fig. 6.9 for the defaults for cattle.
Figure 6.9: Scale function, eqn (6.72), for fat catabolism and partitioning between fat growth
and milk production for a dairy cow.
When this function is negative, there is compulsory fat catabolism, whereas when it is positive it defines
the relative priority for fat growth and milk production. The overall growth dynamics are discussed later.
6.7 Animal intake
Animal intake is defined in relation to feed composition, animal weight and, and pasture availability in the
case of grazed pasture. In the following analysis, all pasture mass and intake units are expressed in carbon
units with the corresponding nitrogen units. In some cases, these will be converted to dry weight or
protein fractions for illustration. However, retaining C and N units in the analysis avoids problems with
conversion factors.
6.7.1 Potential intake
It is common for models to relate potential intake to animal BW in some way. The approach here is to
assume that, for a reference digestibility, normal growth can be satisfied with non-limiting feed availability.
Potential intake at this digestibility is then related to animal body protein, rather than total weight, since,
as discussed in detail above, body protein is taken as an indicator of metabolic state. Thus, for example, if a
mature animal loses body fat but not protein, the potential intake is unchanged.
The approach in the model code is to grow an animal under normal conditions, as discussed above, and
calculate the ME required which is then stored in an array as a function of body protein, ๐๐, so that intake
at reference digestibility is defined by
,pot ref PI W (6.74)
kg d.wt animal-1 d-1. This is then applied at all stages of growth.
During lactation or pregnancy, it is assumed that potential intake increases due to physiological changes in
the rumen. These are considered in turn.
0 100 200 300 400
Days since parturition
-1.0
-0.5
0.0
0.5
1.0
Sca
le fu
nctio
n
DairyMod and the SGS Pasture Model documentation 112
Intake during pregnancy
Potential intake is assumed to increase during pregnancy to provide the extra energy required. The simple
scale function is defined as
1,
,,
preg reqI preg preg
mat req
Ef
E (6.75)
where ๐๐๐๐๐, d, is the time since conception, ๐ธ๐๐๐๐,๐๐๐ MJ d-1, is the energy required for pregnancy, and
๐ธ๐๐๐ก,๐๐๐ MJ d-1, is the energy required at normal mature weight for a non-pregnant and non-lactating
animal. The potential intake during pregnancy is then defined by combining eqns (6.74) and (6.75) as
, , , ,pot ref preg pot ref P I preg pregI I W f (6.76)
Intake during lactation
Potential intake is assumed to increase to a peak following parturition and subsequently decline, according
to the function
1 1, , ,I lact lact I lact mx lactf f (6.77)
where ฮ(๐๐๐๐๐ก) is defined as a normalized gamma function in terms of the time since parturition, ๐๐๐๐๐ก d,
the time to maximum intake, ๐๐๐๐๐ก,๐๐ฅ d, and the curvature coefficient ๐ผ, as given by
1, ,
explact lactlact
lact mx lact mx
(6.78)
where
0 0
1,
lact
lact lact mx
(6.79)
so that
0 1,
, , , ,
I lact lact
I lact lact lact mx I lact mx
f
f f
(6.80)
Note that, in the model, different curvature parameter values for ๐ผ can be used for pre- and post-peak
lactation.
During lactation the potential intake function in eqn (6.74) is scaled according to
, , ,pot ref lact pot ref P lactI I W (6.81)
Default parameter values are
80 0 9 0 5 2 0
80 0 8 0 5 1 8
50 0 8 1 0
dairy: d
cattle: d,
sheep: d,
,, , , , ,
, , , , ,
, , , ,
, . , , . , .
. , , . , , .
. , , . , ,
lact mx lact mx lact mx I lact mx
lact mx lact mx lact mx I lact mx
lact mx lact mx lact mx I lac
f
f
f 2 0, .t mx
(6.82)
where it can be seen that, in this case, different defaults apply to dairy cows and cattle.
If the animal is also pregnant, then the scale function ๐๐ผ,๐๐๐๐ in eqn (6.75) is also applied.
Chapter 6: Animal growth and metabolism 113
Equation (6.78) is illustrated in Fig. 6.10 for the dairy cow default values.
Figure 6.10: Intake scale function during lactation, eqn (6.78)
6.7.2 Intake in relation to feed composition
Animal feed, whether from pasture or supplement, is assumed to comprise three basic components:
neutral detergent fibre (NDF) which is primarily cellulose, hemicellulose and lignin in cell wall material;
protein; and the remainder which is the neutral detergent solubles (NDS) and is mainly sugars for pasture
but may include compounds such as starch and fat for other feeds. The fractions of these are denoted by
๐๐๐ท๐น,๐๐๐๐, ๐๐,๐๐๐๐, ๐๐,๐๐๐๐ respectively so that
1 , , ,NDF feed P feed S feedf f f (6.83)
where subscripts refer to โfibreโ, โproteinโ and โsolublesโ. If these components have digestibilities denoted
by ๐ฟ๐๐ท๐น,๐๐๐๐, ๐ฟ๐,๐๐๐๐, ๐ฟ๐,๐๐๐๐, then the total digestibility is
, , , , , ,feed NDF feed NDF feed P feed P feed S feed S feedf f f (6.84)
It is assumed that, for all feed types, the protein and NDS digestibilities are fixed for all feed types, and are:
0 85 , , .P feed S feed (6.85)
Thus, total digestibility ๐ฟ๐๐๐๐ is influenced primarily by feed composition and the digestibility of the NDF
component.
A digestibility intake scale factor ๐๐ฟ is defined as
,
1
q
feed hmx q
feed h
(6.86)
where
,
1q
ref hmx q
ref h
(6.87)
According to this formulation, the function ๐๐ฟ increases from zero when digestibility is zero, takes values
0.5 and 1 at digestibilities ๐ฟโ and ๐ฟ๐๐๐ respectively, and has asymptote ๐๐ฟ,๐๐ฅ. The coefficient ๐๐ฟ controls
0 100 200 300 400
Days since parturition
0.0
0.5
1.0
1.5
2.0
Sca
le fu
nctio
n
DairyMod and the SGS Pasture Model documentation 114
the curvature of the equation. This equation is based on the general switch type equation discussed in
Chapter 1, section 1.3.3, and is illustrated in Fig. 6.11 for the default parameters.
0 5 0 68 2 . , . ,h ref q (6.88)
Figure 6.11: Intake digestibility scale factor, eqn (6.86), with parameter values in eqn (6.88).
This approach to defining the relative effect of digestibility on intake is applied to any source of intake โ
that is, pasture, concentrate, forage (silage or hay) and mixed ration. NDF is sometimes used directly as an
indicator of the effect of feed quality on intake but I have found that digestibility offers more flexibility and
is simpler to work with mathematically.
It is simple to adjust the shape of the intake response to digestibility by adjusting the parameters in eqns
(6.86) and (6.87). Note that since this function is defined to take the value unity at the reference
digestibility ๐ฟ๐๐๐, it will exceed unity at values above that value.
6.7.3 Pasture intake
Pasture intake depends on availability as well as pasture quality. A scale factor to define intake in relation
to available pasture is defined as
,
paspas
pas ref
(6.89)
with the function ๐๐๐๐ (๐๐๐๐ ) given by
1
,
,
pas
pas
qpas pas h
pas pas qpas pas h
W WW
W W
(6.90)
where ๐๐๐๐ , kg C m-2 is available pasture, ๐๐๐๐ is a curvature coefficient, ๐๐๐๐ ,โ is the available pasture at
which ๐๐๐๐ = 0.5, and ๐๐๐๐ ,๐๐๐ is the value of ๐๐๐๐ at reference pasture ๐๐๐๐ ,๐๐๐ so that
1,pas pas pas refW W (6.91)
When pasture is unlimited, ๐๐๐๐ approaches the limit
0 20 40 60 80
Digestibility, %
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Sca
le fa
cto
r
Intake digestibility scale factor
Chapter 6: Animal growth and metabolism 115
1,
,pas mx
pas pas refW
(6.92)
so that
2
,,
pas mxpas pas pas hW W
(6.93)
Thus, ๐๐๐๐ is readily parameterized in terms of the available pasture at which intake is restricted to 50% of
potential and a reference pasture availability at which the ๐๐๐๐ is unity. Default values for cattle and
sheep, expressed in t d.wt ha-1 (although the model internal units are kg C m-2), are
1
1
0 7 1 5
0 5 1 5
,
,
. , .
. .
pas h pas
pas h pas
W q
W q
cattle: t d.wt ha
sheep: t d.wt ha , (6.94)
and the reference value is
12,pas refW t d.wt ha (6.95)
Thus, pasture intake is
,pas pot ref P pas pas pasI I W W (6.96)
where ๐ฟ๐๐๐ is pasture digestibility and ๐๐๐๐ is pasture availability. Note that if the animal is pregnant
and/or lactating then the scale factors given by eqns (6.76) and (6.78) are implemented.
6.7.4 Supplement intake
Supplement intake is related to supplement composition and digestibility, as well as any management
restrictions in supply. The potential intake is given by the above analysis. However, the intake function in
relation to pasture availability is also applied at its asymptote, eqn (6.92). Thus, the maximum potential
supplement intake is
, , ,mx pas mx pot ref PI I W supp supp (6.97)
where ๐ฟ๐ ๐ข๐๐ is the digestibility of the supplement. Again, if the animal is pregnant and/or lactating then
the scale factors given by eqns (6.76) and (6.78) are implemented.
6.7.5 Substitution
Substitution is the phenomenon where intake of supplement can cause reductions in pasture intake.
Substitution will only occur in response to supplement that is fed prior to pasture. Thus, for example, if a
minimum concentrate is fed followed by pasture, and then more concentrate, substitution is calculated in
relation to the minimum concentrate only. Obviously, total intake is constrained by animal intake capacity.
There is a single substitution coefficient prescribed for the animal on the โStockโ module under the
โBiophysicsโ page. This is the substitution that occurs when pasture availability is non-limiting, and the
default is 0.8.
Once supplement intake has been calculated, the potential pasture intake is then scaled by the function
1
, ,pas mx pas pot ref P
If
I W
supp suppsupp (6.98)
DairyMod and the SGS Pasture Model documentation 116
and, again, eqns (6.76) and (6.78) are implemented if necessary.
6.8 Metabolisable energy and nitrogen dynamics
The theory so far describes the diet composition and digestibility which determine potential animal intake
and I shall now consider calculations for metabolisable energy in relation to feed composition and nitrogen
dynamics, which are central to overall nutrient dynamics in the model. Nitrogen dynamics are considered
first since there are energy costs associated with urine excretion and these affect overall available
metabolisable energy. The net nitrogen balance through the animal is simple in that the input is equal to
the sum of nitrogen retained (body tissue or milk) and that excreted in dung and urine. In the case of a
non-lactating cow maintaining a fixed body weight, excreted N will exactly balance input. However, the
animal requires dietary N to balance turnover of rumen microbes. It is beyond the scope of the present
model to include full rumen functionality, although such models provide valuable insight into rumen
dynamics. A simpler approach will be adopted.
Before proceeding, note that in the following analysis, intake is expressed in d.wt units. Since the model
works with carbon units, care must be taken to ensure that appropriate conversions are made when
implementing the model in code.
Denoting the total intake by ๐ผ, kg d.wt d-1, the corresponding N intake is
,N N P feedI f I (6.99)
where, as discussed above, ๐๐ is the protein fraction, and ๐ผ๐ is the N fraction of protein taken to be
1
0 16.N
kg N kg protein (6.100)
which is equivalent to the usual factor of 6.25 for converting N to protein.
It is assumed that senescence and excretion of rumen microbes is exactly balanced by new growth. It is
further assumed that rumen microbial senescence, ๐ต๐ ๐๐ kg d.wt d-1, is proportional to intake which implies
that as microbial activity increases through a greater intake, so does the turnover of microbes. Thus
sen BB I (6.101)
The corresponding N excretion from rumen microbes is then
, ,sen N B N senB f B (6.102)
where ๐๐ต,๐ kg N (kg d.wt)-1 is the N fraction of the rumen microbes with default value
0 1, .B Nf (6.103)
Now consider dung, ๐ท kg d.wt d-1, which is given by
1
1
sen
B
D I B
I I
(6.104)
and the corresponding dung N, ๐ท๐ kg N d-1 is
1
1
, , ,
, , ,
N N P feed P feed sen N
N P feed P feed B B N
D f I B
f I f I (6.105)
The N fraction of dung can now be written
Chapter 6: Animal growth and metabolism 117
1
1
, , ,,
N P feed P feed B B NN dung
feed B
f ff (6.106)
The default value of ๐๐ต is
0 04.B (6.107)
so that microbial decay is equivalent to 4% of intake.
Thus, for example, for a good quality pasture with digestibility 75%, and 25% protein, the dung N
concentration is 3.4% which is realistic.
Now consider urine N, which is taken to be the excess intake N that is not utilized by the animal or excreted
as dung. The total daily N input balance between intake, retained and losses to dung and urine is
N N N N NI W M D U (6.108)
where ฮ๐๐ is the body weight N balance, which will be negative for protein weight loss, ๐๐ is the N
content of milk (where appropriate), ๐ท๐ is the N loss in dung and ๐๐ the N loss in urine. Thus, ๐๐ becomes
, , ,N N P feed P feed B B N N NU I f f M W (6.109)
This gives the urine N balance in terms of intake, N retained by microbial biomass, milk and body weight
change. For example, if the animal is not lactating and has no weight change, then all digested N that is not
retained by the microbes is excreted as N. The N retained by the microbes is balanced by the
corresponding losses to dung through microbial senescence.
The metabolisable energy available to the animal, ๐ธ๐๐ธ MJ d-1, is the difference between the gross energy of
feed intake and energy costs associated with the production of methane, urine and dung, and can be
written
4 ,ME g CH U NE I E E (6.110)
where ํ๐, MJ kg-1, is the energy density of the feed and the last two terms are energy costs associated with
the production of CH4 and dung respectively, and are considered in turn.
Energy costs associated with CH4 production are assumed to be given by
4 4CH CH gE I (6.111)
where
4 4
4
1 2
,,CH ,
FCH CH ref
F ref
f
f
(6.112)
and ๐๐ถ๐ป4,๐๐๐ is the energy associated with CH4 production as a proportion of digestible energy intake at
reference NDF content, ๐๐น,๐ถ๐ป4,๐๐๐, of the feed. In the model,
4
0 09, .CH ref (6.113)
where
4
0 65, , .F CH reff (6.114)
DairyMod and the SGS Pasture Model documentation 118
Thus, for example, if ๐ฟ๐น = 0.6, ๐ฟ๐ = ๐ฟ๐ = 0.85 then for composition ๐๐น = 0.55, ๐๐ = 0.2, ๐๐ = 0.25, which is
representative of pasture, the fraction of energy lost through methane fermentation is 6.1%, whereas if the
composition is ๐๐น = 0.2, ๐๐ = 0.1, ๐๐ = 0.7, which is representative of a concentrate supplement, the
energy fraction is now 4.2%. These values are consistent with lower energy costs through methane
fermentation for low fibre diets (Beauchemin et al., 2009).
For urine, it is assumed that the energy costs are related to the urine N, so that
, ,U N U N NE U (6.115)
where the urine output, ๐๐, is given by eqn (6.109) and ํ๐,๐, MJ (kg N)-1 is the energy cost of producing N,
with default value
1
30,U N
MJ kg N (6.116)
It is convenient to separate the urine N into the component corresponding to no N retention by the animal
and then allow for any retention as milk or body weight change. Write eqn (6.109) as
0,N N retU U N (6.117)
where
0 , , , ,N N P feed P feed B B NU I f f (6.118)
and
ret N NN M W (6.119)
The metabolisable energy, eqn (6.110), can now be written
,ME U N retE I N (6.120)
where
4
1 , , , ,g feed CH U N N P feed P feed B N Bf f (6.121)
This coefficient, with units MJ kg-1, is termed the apparent metabolisable energy coefficient and is the
metabolisable energy content of the feed in the absence of any N retention by the animal. This means that
the ME available to the animal depends on the N dynamics in the animal and is not a function of the feed
only. For example, if the same feed is given to non-growing animals that are either lactating or not
lactating, the ME content available to the lactating animal will be greater due to the retention of N in milk,
with the difference being due to the costs of excreting surplus N in urine.
6.9 Growth dynamics in response to metabolisable energy
The analysis so far defines the metabolisable energy available to the animal in terms of available pasture
and quality, supplement supplied, and animal metabolic state. It now remains to calculate the overall
growth dynamics including foetal growth and milk production where appropriate. The sequence of
calculations using the theory described above is as follows:
Calculate potential intake from pasture and supplement, including any substitution effects.
Calculate ME required for growth, maintenance and, if relevant, pregnancy.
If the animal is lactating and ๐๐๐๐๐ก < ๐๐๐๐๐ก,๐น0 in eqn (6.72), so that fat catabolism occurs, calculate
the energy released.
Chapter 6: Animal growth and metabolism 119
It then remains to calculate growth. First consider a non-lactating animal in which case, three conditions
are considered.
ME available exceeds requirements for growth, maintenance and pregnancy (if appropriate). In
this case, intake is reduced to maximum requirement and growth is calculated accordingly.
ME available is less than requirements for growth, but there is sufficient fat catabolism to meet
maintenance requirements. In this case there is no growth and the necessary fat catabolism occurs
to provide energy, along with intake, to meet maintenance requirements.
ME available through intake and fat catabolism is insufficient to meet maintenance requirements.
This is an animal that is going to lose weight and does so first through maximum fat catabolism.
Body protein will then be lost as a result of insufficient energy to resynthesis degraded protein
through protein maintenance requirements.
For a lactating animal, lactation will only occur if the energy available exceeds maintenance requirements.
If this is the case and the available energy after meeting other costs is ๐ธ๐๐๐๐ก,๐๐ฃ๐๐๐ and daily milk production
is ๐ฟ, L d-1, then the energy balance is
, , ,M
N M U N lact availM
M f EY
(6.122)
where ๐๐,๐, kg N L-1, is the milk N fraction. This equation accounts for the energy that is available through
N retention in the milk that would otherwise have incurred a cost to be excreted as urine. Thus, the milk
production is
,
, ,
lact avail
MN M U N
M
EM
fY
(6.123)
In practice in the model, milk production will respond to pasture availability and quality as well as feed
management. As a simple example, consider a dairy cow being fed two contrasting feeds:
a fixed good quality pasture based feed with 50% NDF at 60% digestibility, 20% protein so that it is
20% NDS;
a mixed ration feed with 25% NDF at 70% digestibility, 20% protein so that it is 60% NDS.
In both cases protein and NDS have digestibility 85%. The corresponding lactation over 300 days is shown
in Fig. 6.12 where it can be seen that, as expected, substantially greater milk production occurs with the
mixed ration, with total milk production being 5,028 L and 7,271 L respectively. The feed conversion
efficiencies, FCE L (kg feed)-1, are also shown and, as expected these are greatest during early lactation, due
to fat catabolism, and also for the mixed ration. The overall average FCEs are 0.85 and 1.22 for the pasture
and mixed ration fed cows respectively. These examples show two extremes and, in practice, cows are
likely to receive a balance between pasture, concentrate, forage and mixed ration. However, this example
demonstrates the model gives the appropriate response to differing feed supply. It should be noted that
the absolute values for milk production will depend on the size and genetic merit of the cow.
DairyMod and the SGS Pasture Model documentation 120
Figure 6.12: Milk production over 300 day lactation (left) and corresponding feed conversion
efficiency (right) for a cow being fed good quality pasture or mixed ration feed as indicated.
See text for details.
Figure 6.13 shows the body weight and energy dynamics corresponding to the pasture fed cow in Fig. 6.12:
The characteristics of the dynamics are similar for the mixed ration feed supply. The negative growth
energy corresponds to fat catabolism.
Figure 6.13: Body weight (left) and energy dynamics (right) for the pasture fed dairy cow as
illustrated in Fig. 6.12. See text for details.
A detailed comparison between the model and observed experimental data from the Project 3030
โRyegrass Maxโ farmlet in southwest Victoria, Australia has been presented in Johnson et al. (2016). This is
not discussed here other than to note that excellent agreement over 3 lactations was observed for daily
lactation and the corresponding pasture, concentrate and forage intake.
6.10 Final comments
The animal growth and metabolism model has been described that can be applied to sheep, cattle or dairy
cows. It is a generic animal based on Johnson et al. (2012) and Johnson et al. (2016). It includes growth,
pregnancy and lactation, and the full energy dynamics including costs associated with growth of body
protein and fat, resynthesis of degraded protein, termed protein maintenance, maintenance costs
associated with travel, and fat catabolism. Costs of N production through urine excretion and dung are
included. It also includes methane emissions from rumen fermentation and partitioning of N between
0
5
10
15
20
25
30
35
0 100 200 300
Milk
pro
du
ctio
n, L
d-1
Days since calving
0
0.5
1
1.5
2
0 100 200 300
FCE,
L k
g-1
Days since calving
Pasture
MR
0 100 200 300 400
Days since parturition
0
200
400
600
800
We
igh
t, k
g Total
Protein
Water
Fat
Foetus 0 100 200 300 400
Days since parturition
-100
0
100
200
ME
, M
J / d
Total
Lact
Preg
Maint
Grow th
Chapter 6: Animal growth and metabolism 121
dung and urine and so is ideally suited for greenhouse gas dynamics studies. The model is versatile and can
simulate a wide range of pasture and feed management systems.
6.11 References
Agricultural Research Council. (1981). The nutrient requirements of farm livestock, no. 2 ruminants (2nd
ed.) Agricultural Research Council, London.
Baldwin RL, France J, Beever DE, Gill M, Thornley JHM (1987). Metabolism of the lactating dairy cow. III.
Properties of mechanistic models suitable for evaluation of energetic relationships and factors involved
in the partition of nutrients. Journal of Dairy Research. 54, 133-145.
Baldwin RL (1995). Modeling Ruminant Digestion and Metabolism. Chapman & Hall, London.
Beauchemin KA, McAllister TA, McGinn SM (2009). Dietary mitigation of enteric methane from cattle. CAB
Reviews: Perspectives in Agriculture, Veterinary Science, Nutrition and Natural Resources 4, No.035.
Bergen WG (2008). Measuring in vivo intracellular protein degradation rates in animal systems. Journal of
Animal Science, 86, E3-E12.
Dijkstra J (1994). Simulation of the dynamics of protozoa in the rumen. British Journal of Nutrition, 72, 679-
699.
Dijkstra J, Neal HD St. C, Beever DE, France J (1992). Simulation of nutrient digestion, absorption and
outflow in the rumen: model description. Journal of Nutrition, 122, 2239-2256.
Emmans GC (1997). A method to predict the food intake of domestic animals from birth to maturity as a
function of time. Journal of Theoretical Biology, 186, 189-199.
Finlayson JD, Cacho OJ, Bywater AC (1995). A simulation model of grazing sheep: I. Animal growth and
intake. Agricultural Systems, 48, 1-25.
Fox DG, Black JR (1984). A system for predicting body composition and performance of growing cattle.
Journal of Animal Science, 58, 725-739.
Gerrits WJ, Dijkstra J, France J (1997). Description of a model integrating protein and energy metabolism in
pre-ruminant calves. Journal of Nutrition, 127, 1229-1242.
Johnson IR, France J, Thornley JHM, Bell MJ, Eckard RJ (2012). A generic model of growth, energy
metabolism, and body composition for cattle and sheep. Journal of Animal Science, 90, 4741-4751.
Johnson IR, France J, Cullen BR (2016). A model of milk production in lactating dairy cows in relation to
energy and nitrogen dynamics. Journal of Dairy Science, in press.
Lewis RM, Emmans GC (2007). Genetic selection, sex and feeding treatment affect the whole-body
chemical composition of sheep. Animal, 10, 1427-1434.
National Research Council. 2001. Nutrient requirements of dairy cattle, 7th rev. ed.: National Academies
Press, Washington.
Rattray, P. V., W.N. Garrett, N.E. East, and N. Hinman. 1974. Growth, development and composition of the
ovine conceptus and mammary gland during pregnancy. Journal of Animal Science, 38, 613-626.
Thornley JHM, France J (2007). Mathematical Models in Agriculture. CAB International. Wallingford, UK.
Williams CB (2005). Technical Note: A dynamic model to predict the composition of fat-free matter gains in
cattle. Journal of Animal Science, 83, 1262-1266.
DairyMod and the SGS Pasture Model documentation 122
Wright IA, Russell AJF (1984). Partition of fat, body composition and body condition score in mature cows.
Animal Production, 38, 23-32.
Chapter 7: DairyMod management 123
7 DairyMod management
7.1 Introduction
DairyMod provides flexibility in management of stock, stock rotation, feeding options, fertilizer and
irrigation. These routines are described as logical strategies rather than through a detailed mathematical
exposition. It is recommended that this chapter is read while working directly with DairyMod.
7.2 Stock
Only dairy cows can be implemented in DairyMod. Up to 12 sets of animals can be implemented by
defining:
Number of animals
Calving date
Lactation length
Once these have been defined, the animals to be implemented are selected. Note that in the program each
animal class is identified by its โIDโ number โ this is used in the graphs for the simulation output and also if
simulation data are exported to Excel. Consider, for example, a study that wants to compare split versus
single calving dates. In this case the user would do the following:
Define animal ID-1 as 300 animals and set the calving date to that for a single calving
Define animal ID-2 as 150 animals and set the calving date as the first for the split calving
Define animal ID-3 as 150 animals and set the calving date as the second for the split calving
When running the simulations:
Implement ID-1 only to run the single calving system
Implement ID-2 and ID3 only to run the split calving system
Care must be taken to ensure that the total number of animals implemented is appropriate, both for
multiple calving dates and also the total farm size, as defined by the number of paddocks implemented.
7.3 Supplement
Supplementary feed components are:
Concentrate
Mixed ration
Forage
For each of these, parameters that are specified are the NDF (neutral detergent fibre) and protein
percentages, along with the NDF digestibility. The model assumes that all NDS (neutral detergent solubles),
which is everything apart from the NDF, has a digestibility of 85%. Using this information, the model
calculates the total digestibility and ME available, as described in Chapter 6.
7.4 Feed management
Feed management is quite flexible. Four animal metabolic stages are considered, although only those that
are relevant to the stock selection are implemented. The stages are:
Growing
DairyMod and the SGS Pasture Model documentation 124
Mature/empty
Pregnant
Lactating, which may include pregnant
For each of these, any sequence for the following can be prescribed:
Pasture
Minimum concentrate
Maximum concentrate
Minimum mixed ration
Maximum mixed ration
Maximum forage
Obviously not all components need be specified. The only constraints are that maximum concentrate
cannot precede minimum concentrate, and similarly for mixed ration.
A maximum total intake can also be prescribed. This may be useful to avoid overfeeding the animals when
there is plenty of pasture available. For example, if the maximum mixed ration is prescribed to account for
times when there is no pasture available, it may be necessary to limit the mixed ration when pasture intake
is relatively high.
The feeding sequence is prescribed on a monthly basis, although it is easy to copy one sequence to other
months. In the program, the feeding sequences are simply prescribed by โdragging and droppingโ โ this
should be self-evident on the โFeed managementโ tab of the โManagementโ page.
This flexibility in feed management ensures that it is possible to simulate the main feeding regimes used
throughout Australia.
In summary, it is possible to prescribe the feeding sequence for each month of the year and for all
physiological phases of the animal (although they are unlikely to be empty in practical simulations). This
can result in many feed sequence combinations.
Note that care must be taken to ensure the intended feeding strategies are copied correctly between
months.
7.5 Single paddock management
The single paddock management options are for cut or grazing simulations. Click on โGrazeโ under โSingle
paddockโ, which will show the following options:
Cut
Set stocked
Variable stock density
Rotational grazing by pasture weight
Rotational grazing at fixed time interval
These are considered in turn. Note that animal growth will only be implemented for the set stocked option
and with the option to de-stock (see below) not selected. This is because there are times when the animals
are removed from the simulation.
Chapter 7: DairyMod management 125
7.5.1 Cutting simulation
The cutting simulation is particularly useful for running long-term simulations to assess variation in pasture
growth rates. The model has two cutting options:
Monthly cut
Cut by date
With the monthly cut, the paddock is cut to a specified residual on the last day of each month. The average
growth rate for that month is the accumulated dry matter above the residual divided by the number of
days in the month. For example, if the pasture grows from the harvested residual of 1 t ha-1 to 2.8 t ha-1,
then the accumulated dry weight is 1.8 t ha-1, or 1800 kg ha-1. For a month with 30 days, this corresponds
to an average growth rate of 60 kg ha-1 d-1.
Of course, a minor weakness of this scheme is the fact that the number of days per month is variable.
However, by cutting on the last day of each month, it is very simple to analyse long-term simulation data in
a quick and efficient manner. This approach has been used widely in many simulation studies.
For studies that want to work with specific cutting dates, there is also the option to define all cutting dates
on the interface, in a format that is compatible with Excel.
It is important to note that long-term cutting simulations result in considerable dry matter, and therefore
nutrients, being removed from the paddock. Rather than applying unrealistic nitrogen fertilizer amounts, it
is strongly recommended that cutting trial simulations have โInclude soil C and Nโ not selected on the main
program โSimulationโ page and, in this case, the model runs under the assumption of non-limiting nitrogen.
7.5.2 Grazing simulation
There are several options for the single paddock grazing simulation, which are considered in turn.
For these options, animal growth is only implemented for the set-stocked system and when the paddock is
not de-stocked, as described below. This is to avoid having to define a management strategy for the
animals when they are not on the paddock. In these cases, the stock are assumed to be mature, non-
lactating, and non-growing. The main purpose of these options is to focus on the pasture dynamics โ for
milk production it is necessary to define the complete system management.
Set stocked
The set stocked option should be self-explanatory โ the animal numbers prescribed in the โStockโ section
will graze the paddock each day.
There is also an option to de-stock at a prescribed pasture dry weight and then re-stock when the pasture
recovers to another prescribed dry weight. If this option is selected then animal growth is not included and
only mature dry cows will be implemented.
7.5.3 Variable stock density
Variable stock density provides the option to vary the stock numbers in relation to available pasture dry
weight. The interval for stock adjustment is defined, either in days or months. When stock are adjusted,
the difference between actual pasture dry weight and target residual is used to estimate the pasture
carrying capacity which defines the stock numbers until the next time of stock adjustment. There is also
the option to prescribe a maximum stock density.
DairyMod and the SGS Pasture Model documentation 126
The main purpose of this management option is to assess the variability in the stock carrying capacity of the
paddock. It is not immediately of practical value, although it does give useful insight into the inherent
variation in pasture systems.
7.5.4 Rotational grazing by pasture dry weight
According to this scheme, stock remain on the pasture until it falls below a specified dry weight and are
then returned to the pasture when it reaches another specified dry weight. Stock numbers remain fixed at
the value defined for wethers or steers, depending on the stock choice of sheep or cattle.
7.5.5 Rotational grazing at fixed time interval
This is a simple approach for looking at a single paddock that is being grazed at regular intervals. The time
between grazings is specified and then the stock spend one day on the paddock. The paddock will not be
grazed below the prescribed residual.
7.5.6 Rotational grazing by feed on offer and days on paddock
According to this strategy, the dry weight at which to graze the paddock is defined, along with the target
residual dry weight and number of grazing days. The actual number of animals is then calculated from
animal intake requirement to ensure the paddock is grazed to the target residual in the specified number of
days.
7.5.7 Rotational grazing by date
The final grazing option allows the date and number of animals for each grazing event to be specified
individually. The form for entering the data is compatible with Excel.
7.6 Multiple paddock management
The primary paddock management system for dairy farms is intensive rotational grazing. There are three
rotational grazing strategies that can be implemented in DairyMod:
Fixed time rotation
Rotate in relation to pasture dry weight
Rotate in relation to leaf stage and pasture dry weight
Note that for some of the systems described above, a โholding paddockโ is used, whereby the stock are
moved to the holding paddock under particular circumstances. This is always paddock 1 in the model.
The rotational grazing strategies are considered in turn.
7.6.1 Fixed time rotation
This is a simple rotation scheme that moves the stock around the paddocks, leaving them on the paddock
being grazed for a specified number of days. There is the option to move them in sequence, in which case
they rotate around the paddocks regardless of dry weight. If this option is not selected, then the stock
move to the paddock with the greatest dry weight.
7.6.2 Rotate in relation to pasture dry weight
According to this scheme, a paddock becomes eligible for grazing when it reaches a specified dry weight.
Two residuals are defined for the paddock being grazed: the first is the minimum residual, so that the
paddock wonโt be grazed below this value; and the second is the residual at which stock will be moved.
Chapter 7: DairyMod management 127
Thus, stock will remain on the paddock until it reaches the defined residual at which they will be moved.
Having a residual that defines when stock will move that is different to the minimum residual at which the
paddock will be grazed allows the option of moving stock before pasture availability becomes too low.
When stock move, the paddock that they move to is the one with the highest dry weight, provided it is
eligible for grazing. If no paddock satisfies these criteria then a second set of rules apply, whereby a critical
dry weight is defined, along with a holding paddock. If at least one paddock is above the critical dry weight
then you can select between the following options:
Move to holding paddock. In this case, the stock are moved to the specified holding paddock.
Stay put. The stock are not moved and continue to graze the paddock they are already on.
Chase the feed. They move to the paddock with the highest dry weight. Because at least one
paddock exceeds the critical dry weight, they will therefore be moved to a paddock with at least
this amount of pasture.
Thus, if no paddock is eligible for grazing, the options are either to move to the holding paddock or stay
put, with the further option of chasing the feed if at least one paddock is above the critical dry weight.
7.6.3 Rotate in relation to leaf stage and pasture dry weight
The management strategy of moving stock in relation to leaf stage is quite widely used in dairy systems. It
is similar to implement to 7.6.2 apart from the fact that paddocks are deemed eligible to be grazed if either
the leaf stage or pasture dry weight exceed prescribed values. The rest of the rules are identical to the
scheme for rotating in relation to pasture dry weight.
Note that for pastures with multiple species, leaf stage is defined as the weighted average in relation to
pasture dry weight.
7.6.4 Cutting rules
For each of the rotational grazing strategies a set of cutting rules can also be implemented. The pasture dry
weight at which to allocate the paddock to cutting is defined. The paddock is then removed from the
rotation and subsequently cut either when it reaches a prescribed dry weight or after a defined number of
days since it was allocated to be cut, whichever occurs first. We use number of days and not just a dry
weight because, if growth slows for a reason such as lack of rainfall, then it is possible that the paddock will
never reach the dry weight required for cutting. The cutting rules can be applied to any paddocks.
If paddocks have been allocated to be cut and there are no paddocks eligible to be grazed, they will be
returned to the grazing cycle. If this is necessary, the paddocks that have been allocated to be cut with the
lowest dry weight are returned to grazing first. This means that paddocks that have recently been removed
from the grazing cycle are preferred over those that are almost ready to be cut.
7.6.5 Rotational grazing phases
In many enterprises, different rotational grazing strategies are used at different times of year. For example,
during summer it may be in relation to pasture dry weight, whereas in spring it may be according to leaf
stage. The model allows 4 distinct phases to be implemented, each with its own starting date.
7.6.6 Crops
DairyMod also includes the option to implement forage crops in the simulation. Cereals, brassicas and bulb
crops can be implemented. For the cereals, both C3 (eg wheat) and C4 (eg maize) can be prescribed. The
crop growth is described in detail in Chapter 3. Double cropping is available, as described below. The
DairyMod and the SGS Pasture Model documentation 128
grazing rules for cereals are different to those for brassicas and bulb crops. Note that a specific crop
paddock is selected by the user, and crops will only be grown on this paddock.
Cereals
For cereals, the following rules apply:
Crops are grazed as normal based on dry weight criteria, with the following specified:
o Dry weight at which the crop is to be grazed, default 3 t ha-1
o Minimum graze residual, default 1 t ha-1
o Residual at which to move the stock, default 2 t ha-1
o Note that if the crop is eligible to be grazed this will take precedence over other paddocks
The user can select whether to graze the crop at boot phase. If this is selected, following grazing,
the crop is finished and will either be sown back to pasture or to the second crop if selected (see
below)
If the crop is not grazed at booting, then it will be grown out to harvest. This can either be at the
soft dough stage, suitable for silage, or at maturity, suitable for grain. If โCut at soft dough phaseโ is
not selected, it is grown to maturity.
Brassicas and bulb crops
The brassica and bulb crops that are included are rape, kale and turnip. The phases considered for
brassicas are simplified in comparison to cereals, with just vegetative to anthesis and anthesis to maturity.
The growth characteristics of rape and kale are very similar with both being leafy plants that are best grown
while vegetative but, if allowed to flower, resulting in a high proportion of stem material and poor quality
feed. Both kale and rape require vernalization for flowering and so, in practice, are generally sown in spring
or summer to be grazed as a vegetative crop to fill a feed gap. Consequently, while the model does include
vernalization parameters these are generally not relevant in practical situations. The main difference
between kale and rape included in the default parameter sets is that the stems in rape have a lower N
content and therefore nutritive value. The growth characteristics for turnip differ from kale and rape in
that there is no vernalization requirement and bulb initiation and growth occurs.
Brassicas and bulb crops are only grazed once, and the time of grazing is prescribed as the number of days
since emergence. They are then grazed to a prescribed residual before the paddock is re-sown.
Single- and double-cropping
The model allows both single- and double-cropping. The initial crop is sown at the date specified by the
user, and care should be taken to ensure this is appropriate. For example, spring wheat will generally be
sown at a later date than winter wheat. In the case of double-cropping, the date specified for the start of
the second crop is ignored and this crop is sown on completion of the first crop. Note that it is possible that
the first crop will never grow to completion so that the second crop is not implemented โ this may happen,
for example, if maize is implemented to grow to maturity in a cool region and the growth cycle is not
completed in 12 months.
7.7 Nutrient removal
Nutrient returns as dung and urine play an important role in system behaviour through their influence on
nutrient dynamics and the corresponding plant response. In many practical situations, such as dairy
enterprises, substantial amounts of nutrients may be removed from the paddock through dung and urine
Chapter 7: DairyMod management 129
deposition in laneways and the milking shed. The model allows you to define a fixed proportion of dung
and urine nutrient removal from paddocks.
7.8 Nitrogen fertilizer
Up to 4 nitrogen fertilizer application rules can be implemented, and these can be applied to any paddocks.
Combinations of these rules can be implemented. The amount of nitrogen applied can be specified as
nitrate, urea / ammonium, or a combination of both. If urea is applied, it is assumed to be immediately
hydrolysed to ammonium.
Critical soil N (ppm). The nitrogen is applied when the soil N concentration in the top 15 cm falls
below a prescribed value. (Note that ppm is equivalent to mg kg-1)
Plant N as % of optimum. The nitrogen is applied when N concentration in live leaf material falls
below a prescribed percentage of its optimum.
Rotation (cut or grazed). For rotationally grazed systems, or the single paddock cut simulation,
nitrogen is applied after each grazing or cut regardless of soil or plant nitrogen status. For this
strategy, it is also possible to implement fertilizer N as a daily equivalent. Thus, for example, if the
fertilizer amount is defined as 1 kg N ha-1 and the regrowth period prior to grazing or cutting was 35
days, then 35 kg N will be applied following grazing or cutting.
Fixed date. Fertilizer N is applied on a specified fixed date for each year.
Note that:
For the first two strategies there is also a minimum number of days between applications (although
this can be set to zero).
For the first three strategies, fertilizer application can be restricted to a date range. This is useful,
for example, to avoid application during hot dry summer periods in winter dominant rainfall
regions such as WA or Victoria.
7.9 Irrigation
Up to 4 irrigation strategies can be implemented, and these can be applied to any paddocks. As for
fertilizer, combinations of these strategies can be applied. For each strategy, the amount of water as well
as the start and end time are specified. The time and duration of application can affect evapotranspiration
and possible runoff. In addition, irrigation can be limited to a prescribed date range.
Rainfall deficit. The cumulative deficit between potential evapotranspiration (PET) evaluated
according to the FAO56 method and rainfall is calculated each day. Irrigation is applied when this
deficit reaches a specified value. If the deficit falls below zero, which will occur when there is
significant rainfall, it is reset to zero.
Soil water status. Plant available soil water is calculated in the root zone and irrigation is applied
when this falls below a prescribed percentage of field capacity.
Plant water status. Irrigation is applied when plant transpiration falls below a specified percentage
of demand.
Fixed intervals. Irrigation is applied at prescribed fixed daily intervals regardless of any other
considerations.
DairyMod and the SGS Pasture Model documentation 130
7.10 Final comments
The management routines in DairyMod give users considerable flexibility in setting up complex dairy
management systems. In addition, it is possible to analyse single paddocks where the animal responses are
simplified, so that animal growth is not considered and the primary role of the animals is to graze the
pasture and return nutrients through dung and urine. These single paddock simulations play an important
role in helping our understanding of the complex biophysical dynamics in paddocks.
For multi-paddock rotational grazed systems, the model has a range of options that allow simulation of
realistic grazing systems in Australia and elsewhere. These rules are complex and continue to evolve with
the input of key users and specific projects. Indeed, the rules have resulted from close collaboration
between researchers and advisors in order to give a clear reflection of grazing management in Australian
dairy enterprises.